
Spaces of Endomorphisms on PseudoOrthogonal Vector Spaces  Given a real pseudoorthogonal vector space (VV,<,>)of dimension n, i.e. <,> a nondegenerate symmetric bilinear form, as in [ T78 ] and [T84], the automorphism or pseudoorthogonal group of the pseudoorthogonal vector space isits derivation or pseudoorthogonal Lie algebra ( with respect to the commutator ) isThey are related via the standard exponential series for endomorphisms, exponentiating elements of the Lie algebra to elements of the group. A typical, generating linearly element isfor u ∈ VV,giving the typical commutation relations basisfreewith respect to the commutator [ , ]−  note that these also can be found for 2nd power polynomials in its Clifford algebra. Its antiderivation or pseudoorthogonal Jordan algebra with respect to the anticommutator is given by the vector space of the <,>selfadjoint endomorphismswith the generating linearly standard elementand the defining, rather simple anticommutation relationswhich by polarizing twice ( or directly ) give the more general formThey give rise to the hope to find a corresponding composition law for the symmetric space on the set of invertible selfadjoint endomophisms. However, in general not all elements in the space of selfadjoint endomorphims are of this form, they are only finite sums ( the number of terms of which being between 1 and the dimension n ) of these typical generating elements o+x,y. There is a direct and <,>orthogonal decomposition ⊞ of this Jordan algebra, given by
 1   1 
A =  —  trace(A) idVV ⊞ ( A −  —  trace(A) idVV) , 
 n   n 
in a direct sum of scalar multiples of the identity and the traceless endomorphisms, which are embedded as the first two subspaces in the decomposition of the space of curvature structures below. In case of traceless endomorphisms, or in the group case of elements with determinant 1, we write the letter s or S in front.
Exponentiation leads to this fourth category of symmetric spaces in the sense of Ottmar Loos, the local (tangent) structure of which is exactly the real part of the Jordan algebra, studied for Hilbert spaces in ipmWeylP. The pseudoorthogonal vector space (VV,<,>)decomposes into the disjoint union of the two subsetswhich are not subspaces. The first is a flat open subspace of dimension n, the second a connected curved cone of dimension n1. This decomposition is left invariant under the action of the pseudoorthogonal group.  ℝ are the real numbers
these are the classical categories on pseudoorthogonal vector spaces
throughout this article <,>skewadjoint means
<Ex,y> = − <x,Ey>
for any vectors x, y and endomorphisms A, E and
<,>adjoint
means
<Ax,y> = <x,Ay> 
Pseudo Orthogonal
Involutions and Reflections  For a number ઠ from the groundfield and a given vector t outside the nullcone define the linear transformation ઠᦔt on this vector space by
   <t,u>     ઠ 
ઠᦔt u  =  u + ઠ 
 t  which is  ( idVV+ 
 o+t,t) u 
   <t,t>     <t,t> 
in obvious notations: With • in place of the running vector from the underlying vector space, clearly
 <t,•> 
ઠᦔt ○ઠᦔt = idVV+ ઠ (ઠ+2) 
 t  and  ઠᦔλt = ઠᦔt  
 <t,t> 
for a nonvanishing λ from the groundfield, i.e. the ᦔ's are not linear but homogenous of degree 0 in t . This means that the vectors of a whole ray ℝt map onto the same reflection. For any such vector t we get the
Structure Theorem of Involutions and Reflections
Such an endomorphism ઠᦔt is <>selfadjoint, i.e. in this Jordan algebra of endomorphisms and therefore never in the pseudoorthogonal Lie algebra. In the case of involutions we drop the ઠ, writing
   <t,u>     2 
ᦔt u  =  u − 2 
 t  =  ( idVV− 
 o+t,t) u  , 
   <t,t>     <t,t> 
which reflects any element of the onedimensional span of t at its <>perpendicular n1dimensional complementary subspace, i.e. for any λ of the groundfieldholds. It is the special case s = ± t of the more general <,>selfadjoint, but in general not pseudoorthogonal endomorphism
 2   1 
ᦔt,s = idVV− 
 o+t,s i.e. simply ᦔt,su = u − 
 ( <t,u>s + <s,u>t ) 
 <t,s>   <t,s> 
if t and s are not <,>perpendicular. Sometimes the group generated by reflections is called the Weyl group of the pseudoorthogonal vector space [ L&N p 362], not to be confused with the Weyl group whose Lie algebra is the Heisenberg Lie algebra or else the group of pseudoorthogonal transformations plus dilatations. For reflections we get the fundamental formula[L&N p 351]. Reflections are not a group, but (ff) means, that they are a symmetric subspace with respect to the Loosmultiplication ₪ on the group ( and hence symmetric space ) of all invertible linear transformations and especially the pseudoorthogonal group, seen as a symmetric space. As an involution ᦔt has the associate involution − ᦔt and hence the projector ⼫t
     <t,•> 
 ᦔt  ⇄  ⼫t  = 
 t 
  association    <t,t> 
complementation  ⇅   ⇅ 
     <t,•> 
 − ᦔt  ⇄  idVV− ⼫t  = idVV− 
 t   . 
     <t,t>  
Obviously ⼫t projects onto the onedimensional span of t and its complementary projector onto the <,>perpendicular n−1dimensional subspace.
Since all these transformations are linear, they are (outer) automorphisms of the additive abelian group ( VV, + )of the underlying vector space. To get Loos' formulation of a symmetric space we therefore can take æ = ᦔt for involution. Then the fixed point group or generalized unitary group becomes the set of vectors x with <t,x> = 0, which is the orthogonal complement of the linear span of t, and the set of fixed points becomes this linear span itself. In this flat case both subsets are symmetric spaces with respect to the Loos multiplication, which becomes according to (g₪) in the general theory thereonsince we are in the additive abelian group of a vector space. Problem: Find involutions and projectors, which are not of this reflection type, even not reflections at a more than onedimensional subspace. And remember: This only works since t is not lightlike, i.e. not on the light– or more general not on the null–cone.  play an essential role in the general theory of root systems 
Algebra and Geometry
of
Pseudo Orthogonal Vector Spaces  A simple and fundamental algebra structure on such a pseudoorthogonal vector space is the Jordan algebra compositionfor any t ∈ VV,special case of those described on this website on Hilbert spaces for use in quantization. This left multiplication is the sum of a <,>symmetric and a <,>skew endomorphismwhich facilitates many calculations. The two symmetric bilinear forms of this Jordan algebra are the canonicalone, which is nondegenerate for n > 1, i.e. a nontrivial case, and the always nondegenerateone, which is used in the structure theory of Jordan algebras. Contrary to Lie algebras the left multiplications L▣(x) are not derivations ( or antiderivations ). But sums of their commutators are, the socalled inner derivations of the Jordan algebra. Exercise: Dropping the last term in this composition and changing the plus to a minus sign, do we get a nontrivial ( if so nilpotent ) Lie algebra? And what for symplectic vector spaces? For t ∈ ※(VV,<,>)there is the neutral elementof this Jordan algebra (VV, ▣t)and the inverseThe space Inv(VV, ▣)of invertible elements of a Jordan algebra, which here is the outside of the nullcone ※(VV,<,>) ,has a symmetric space composition ₪, which here is
  <x,y>   <x,x> 
(o₪)  x ₪ y = 2 
 x − 
 y  ( especially  x−1 = e ₪ x  ) , 
  <y,y>   <y,y> 
which even is defined for a vector x on the nullcone ⛌(VV,<,>).Note how the symmetric composition o₪ generalizes the inverse, which is a special case of more general negative powers in the symmetric space and in the Jordan algebra together, since powers in both structures coincide if suitably defined ( see below – everything taken from the books of O. Loos ). Obviously
  <x,e>   <x,x> 
e ₪ e = e , e−1 = e but x−1 = e ₪ x ≠ x ₪ e  = 2 
 x − 
 e , 
  <e,e>   <e,e> 
which shows that the multiplication ₪ is not commutative. Unlike in associative categories i.e. inversion is an automorphism of the symmetric space like all leftmultiplications with an element of the vector space. The left multiplication Sx in a symmetric space is given by Because of the 3rd symmetric space axiom it is an ( socalled inner ) automorphism of the symmetric space, here ※(VV,<,>),because of the 2nd it even is involutive. Hence the 1st three symmetric space axioms state that every point is a fixed point of an involutive automorphism and the 4th axiom states that it is isolated. Application to x resp. −x shows, that there is no vector x outside the nullcone such that its inner automorphism is the identity map on ※. Compatability of <,> and ₪ reads
  x   y 
<Su(x) , Su(y)> = <u,u>²  < 
 , 
 >  , 
  <x,x>   <y,y> 
i.e. we get a nonlinear transformation of conformal type outside the nullcone. The symmetric space (※,₪) admits the Jordan algebra (VV, ▣t)as its (local) tangent structure, corresponding to the Lie algebra as the (local) tangent structure for a Lie group.
Loos has shown in his second chapter, that defining for any Jordan algebra ( where we drop the dependency on the neutral element e on the left hand side ) there is no danger of confusion with the quadratic representation of the Jordan algebra since they coincide outside the nullcone. The Ƥ(x) generate a normal subgroup of the automorphism group of the symmetric space Inv(VV, ▣)of invertible elements for any Jordan algebra with a neutral element. It is called the group of displacements because it reduces to translations in the flat case [ Loo p 66]. Its dimension is n, that of the vector space VVof the Jordan algebra . Since the Sx are homogeneous of degree −1 in the 2nd variable these Ƥ(x)transformations turn out to be linear with
 <u,u> 
<Ƥ(u) x , Ƥ(u) y> = ( 
 )² <x,y> 
 <e,e> 
describing the compatibility of <,> and displacements as a conformal property. The 2nd symmetric space axiom identifies the neutral displacement by the inverse in the displacement group for x ∈ VVbeing This easily is proved [Loo p 72] by applying Ƥ to y = Ƥ(y) y−1 in the fundamental formulafor Jordan algebras. Substituting herein y−1 for y and remembering that Ƥ(x) y−1 = x ₪ yresults, where the symmetric multiplication on the right hand side is the usual f ₪ g = f g−1f one in associative structures. This means that the fundamental formula of the Jordan algebra gives as the adjoint  or better self representation of the symmetric space: The set Inv(VV, ▣)of invertible elements, here the outside of the nullcone, thus turns out to be a representation space of the symmetric multiplication. However, it is not defined on the nullcone! Another version of the the selfrepresentation is if we insert the symmetric space definition of Ƥ into the fundamental formula to get dividing by the involutive inversion Se. More general powers are given in a Jordan algebra with unity element, which is ( not associative but ) but powerassociative, in the ordinary way: Defining they coincide with ₪powers: Using induction where 2m−n is the symmetric multiplication of an additive abelian group. Herein the first equation gives the possibility to define an exponential series like in associative algebras with This exponential series maps vectors into vectors in the set of invertible elements. One expects that exponentials ₪generate the connectivity component of e. The exponential series allows the determination of an infinitesimal transformtionof the ₪left multiplication Sx if we expand this nonlinear transformation as with eτ x = e + τ x + terms with higher powers of τ and collect the terms which are linear in τ to get
  2 
u ι➥ ℓíℯSx(u) =  ( <e,u>x + <x,u>e − <e,x>u ) 

  <u,u> 
with the bracket the linear transformation 2 o+x,e−<x,e> id on VVwith trace (1−n)<x,e> and a dilatation. The infinitesimal transformation of the quadratic representation follows by rightcomposing with Se, i.e. by applying the infinitesimal left multiplication to the inverse, to give i.e. the infinitesimal quadratic representation of a Jordan algebra is the left multiplication. In this example it becomes after a verificationThus commutators of infinitesimal left multiplications generate the Lie algebra of the infinitesimal displacements, but themselves they admit no simple composition law in the general case. Expanding in (o₪) both variables x,y as exponential series [ Loo p 95] and collecting the terms linear in the scalar parameter of this series, we get the additive abelian symmetric space composition ₪a
(₪a)u ₪a v = 2 u − v ,
which is abelian in the sense of symmetric spaces [Loo p 134]. These straightforward calculations also work in the more general complex unitary case. Is there on the null or lightcone taken as underlying manifold of dimension n−1, instead of the ndimensional space of invertible elements, a similar symmetric space composition ?  typically s,t,u,v,w,x,y,z are vectors of VV
l even in the nullcone ⛌( , )
wherein we frequently drop arguments
capital letters A,B, ... are reserved for linear transformations on VV
for nonlinear transformations we use script letters
those are nonclassical categories
☞  general Jordan algebras in the box  ☜ 
For symmetric spaces the quadratic representation corresponds to the adjoint representation of associative and Lie structures ! i.e. a map
Ad : ※ → Aut (※,₪)
subject to
Ad(x ₪ y) = Ad(x) ₪ Ad(y)
= Ad(x) ○ Ad(y)−1○ Ad(x)
displacements more useful in
relativity and quantum mechanics
than translations ? are there similiar constructions for < t , t > = 0 ? giving rise to a structure theory of photons 
Null or LightCones  Both subsets ※ and ⛌ inherit their topological structure from that one of VV,but this must be made compatible with <,>, which is standard only in the positivedefinite case. The more interesting and underinvestigated second subset is a cone of dimension n−1, but not convex. The first one is an open subset, hence of dimension n. If <,> is positivedefinite, the nullcone becomes the zero vector 0 only, and the first one a punctured plane. For indefinite <,> both are noncompact. The nullcone has one connectivity component, its complementary set outside the nullcone has two, three or four connectivity components, depending on dimension and signature. The invertibility of reflections results in the
 Lemma
Geometry of NullCones  Two nonvanishing vectors Ĭ and Ĭ' in the nullcone can be reflected into each other by the reflection J Ĭ + Ĭ' if they are not <,>perpendicular. Reflections hence act transitively on nullcones and
ᦔ : t ι➥ ᦔ t , ᦔ : ※(VV,<,>) → ᦔ※
defines a bijection of the symmetric space of the vec tors outside the nullcone ※ into the symmetric space of reflections on ⛌. 
For the proof note that nullcones neither are subspaces nor convex ■ This makes both disjoint sets representation spaces of the symmetric space of all invertible elements of our Jordan algebra, both not vector spaces. But note – this does not make the nullcone itself a symmetric space, which remains an open question. For Ĭ on the nullcone we get a nonlinear map
 <Ĭ,y> 
SĬ : y ι➥ 2 
 Ĭ  ,  SĬ : ※(VV,<,>) → ⛌(VV,<,>) 
 <y,y> 
from outside onto the nullcone, which is not defined only on the nullcone itself.  how to find a <>compatible topoloy ?
reflections belong to the category of symmetric spaces 
Curvature on SpaceTime  The real highdimensional vector space curv(VV,<,>)of pseudoRiemannian curvature structures is defined by bilinear mapssubject for any x,y,v,z ∈ VVto Singer & Thorpe's three axioms [ST]on a real pseudoorthogonal vector space ( VV,<,>),especially for physical Minkowski space, those of dimension 4 and Minkowski signature [ Nom ], [T84] ( and many more publications ). Note that the second axiom sometimes is substituted for another equivalent one. There always is thewhich defines a 1dimensional subspace in the curvature space, called the space of scalar curvature. It will be used below to give three classes of examples. Note that ½ Ro(x,y) = oy,x , but the Bianchi identity does not reduce to the Jacobi identity for Ro.
 this axiomatic has been overlooked in physics 
The Structure Theory of Curvature  Mathematical data, in terms of which the structure theory of curvature spaces can be formulated, are constructed in terms of basic linear algebraic structures: Theis a symmetric bilinear form, which may be used to define R semisimple if it is nondegenerate and compact if it is positive or negativedefinite. Since <,> is nondegenerate Witt's theorem gives a unique ( necessarily <>selfadjoint ) endomorphism on VV ,theand a linear form in the dual space VV*,theIn addition we need an endomorphism, thewith an injective linearespecially Ω(idVV)= Ro , and the nonlinearMoreover there are two projectors on curv(VV,<,>),thewhere idempotence and moreover commutatability are easily verifiedsee [ T78 sec 3] for a more complete list of formulas and a commutative diagram of short exact sequences, visualizing the following structure decomposition. We get [ ST ]
Singer & Thorpe's Structure Theorem of Curvature Spaces wherefrom we get the direct vector space decomposition of type kernШ ⊞ imageШ ofuniquely decomposing every curvature structure into a first cosmological part kern(ℇ−Ш) and a second Weyl part. This should be compared to the decomposition of every electromagnetic field into a stationary and a wave part. The 1st class of more general examples is given for any t ∈ ( VV,<,>)bywhich, as was shown in [T84], comes from a Jordan algebra, in fact that on a real pseudoorthogonal vector space, which was studied elsewhere on this webpage and is basic for the quantization of curved spacetimes. It is constructed for the Jordan composition ▣t asSince it is quadratic in t polarization t ι➥t+s gives even more general curvature structuresNote that since the last term herein is a curvature structure the ½( )sum also is one. ObviouslyBecause the last equation is square in t it can be polarized t ι➥t+s towhich can be easily verified directly. This sorrow result shows that we do not get access to the space of Weyl curvatures by the two vector parameters only. We call curvature structures of this type weak, they take place in kernШ only. The sectional curvature of these curvature structures turn out not to be constant, i.e. they depend on x,y. The other data of these t,smodifications of Ro areIn [T84] there is a more detailed wellknown construction of curvature structures from semisimple Lie and Jordan algebras and Lie triples. The 2nd class is given for any pair A,B of <,>selfadjoint endomorphims on VVbydefining elements RoA,B in the curvature space. The special case B = idVVof which was used above to construct the map Ω. We call curvature structures of this type strong. There is linearity in the variables A,B,and for C = LRoA,Bwith a sorrow consequence: No insight into imageШ by this 2nd class of examples. In fact Ω is a linear bijectionwhich carries over any structure from the ( Jordan algebra of ) selfadjoint transformations into the curvature space. The full diagram of short exact sequences was given in [ T78 p 1120]. The data of this A,Bmodification of Ro are( reproducing the canonical bilinear form ≪,≫ of endomorphisms ) and especiallyAs only chance to look inside imageШ remains Nomizu's construction of
the 3rd class of curvature structures ИE,D , defined asin terms of two <,>skewadjoint endomorphisms E and D, similar but not identical to the 2nd class construction of curvature in terms of two <,>selfadjoint endomorphisms. We call curvature structures of this type em. The data of this E,Dmodification of Ro areThere remains to show that and how Nomizu's construction leads from a physical electromagnetic field, given by electromagnetic field strength E and intensity D, to a gravitational field, solving Einsteins field equations in terms of a ( but which? ) electromagnetic energymomentum tensor. Here E combines the electric field strength E and the magnetic field strength H, whereas D is the electric intensity D combined with the magnetic intensity B. Looking for more curvature structures there are besides И only two more solutions, if one supposes bilinearity in the two parameters E,Dwhere the anticommutator of two <,>skewadjoint endomorphisms is <,>selfadjoint.
A 4th ,mixed' class of curvature structures is given by one <,>selfadjoint A and one <,>−skewadjoint E. If they are supposed to be linear in their two parameters, then [ T84 p 14] there are only the two caseswhere • is the coadjoint action by commutators, and they lie in the first two subspaces of the above curvature space decomposition.
Note that these curvature structures must be linearcombinations of elements of the pseudoorthogonal Lie algebra. In [T78] and [T84] Lie group and algebra actions on these classes of examples of curvature structures were given, together with their data.  Singer & Thorpe curvature structure theory  a great step for the mankind up from linear to multilinear algebraprojectors are a great mathematical achievement,
wellknown as quantum mechanics' statistical operators
but never used in gravity and the symplectic phasespace formulation of classical mechanics
why? 
In and Covariance Groups  Chasing the pseudoorthogonal groups and Lie algebras on (VV,<,>)around was done in [T78], where even dilatationsare included to a larger linear transformation group GGof dimension 1+½ n(n−1), called the linear conformal or sometimes Weyl group of the pseudoorthogonal vector space ( not to be confused with the Weyl group in the classification of real simple Lie algebras ). It is straightforeward to chase the group and Lie algebra • actions around the diagrams and decompositions once they are defined in the wellknown Ad, ad way. Blowing up the pseudoorthogonal group is not necessary, but it collects orbits of diffeomorphic shape into one. Moreover it is easy to check that GGis a subgroup of linear transformations in the automorphism group of the symmetric space (※,₪). Since o₪ is not linear in its two variables, infact it is homogeneous of degree 2 in x and of degree −1 in y, this automorphism group need not be a linear group. Loos defines the group of displacements for any Jordan algebra as a normal subgroup in the automorphism group by left multiplications, which results in nonlinear transformations, in this special symmetric space in square ones in the left variable. The direct decomposition of the curvature space commutes with these Ad, adactions, i.e. is invariant, hence the notation ⊞. Therefore the classification of group orbits takes place inside the three vector space components of the curvature space seperately. Inside kernΩ it is traced back to the classification of GGorbits inoo+( , ) ,one of the classical real simple Jordan algebras of endomorphisms, but inside imageШ it is completely unknown and probably more involved as that unsolved one of nilpotent and solvable Lie algebras. A structure theory usually means that there is a decomposition into ( the direct sum of ) subspaces in terms of projectors, like the above ⊞Singer & Thorpe's one of all curvatures, such that there is a structure theorem. A classification by orbits is a refinement of such a structure theory with the help of a transformation group, if such a group exists and if it commutes with the projectors onto the subspaces.  including dilatations by a real factor λ chases this around formulas and diagrams 
Einstein's Gravitational Field Equations  Einsteins gravitational field equations without cosmological term were given in our notation in [ T78 p 1123] and [T84] as
 Sc(n) 
LR − 2 
 idVV  =  ⓖ TR 
 n 
with ⓖ the gravitational constant and T ∈ o+(<,>) the energy momentum transformation. ⓖ is an universal constant  attempts by Jordan and Dirac in the early days of general relativity to assume a timely decrease in order to explain the movement of continents on earth and their drift could not be verified experimentally so far. Physically T is given by the distribution of sources of the gravitational field, mathematically it should be supplied with an index R. There also is the shorter formone, which corresponds to lowring or highring indices in the index notation. Also from Sc(R) = − ⓖ trace TR which in some cases is easier to handle.  n = 4 and the the signature +,−,−,− is the special case of general relativity 
Gravitational and Electromagnetic Radiation  Gravitational waves, first postulated by Henri Poincaré 1905 in the context of special relativity [ Poi p 1507], and described mathematically by André Lichnérowics [ Lic p 45], are defined for any x,y,z ∈ VVin terms of a (necessarily) lightlike ( for indefinite <,> this means on the nullspace ) vector l ∈ VVwithIf this is the case R is called a gravitational wave and l its wave vector. This is an orbit property with respect to the pseudoorthogonal group, even with respect to its Weyl group. For R a gravitational wave with respect to the wave vector l, the main theorem
 ρR(x,y) = τ <x,l><y,l> for some real τ, i.e. ρR is degenerate and R not semisimple,
 LR = τ o+l,l ,
 Sc(R) = 0 ( i.e. R ∈ image(ℇ−Ш) ⊞ imageШ ), trivial curvatures of ℝRo cannot be waves,
 l is an eigenvector of eigenvalue 0 for LR
on gravitational waves was proven in [T84]. Hence Ω(o+(l,l)) are examples of gravitational waves in image(ℇ−Ш) and even, and as it was shown there also, the only ones in this subspace of the curvature space. Like the curvature structures in this subspace are parametrized completely by <,>selfadjoint endomorphims, gravitational waves therein are parametrized by lightlike vectors, both via the linear map Ω. Candidates of physical gravitational wave orbits are those special wave orbits which are not of electromagnetic origin, because the latter are not strong enough to be measured and they can be ,seen'. To experimentally verify the existence of gravitational waves it is necessary to get an idea of the frequency resp. wavelength, both defined in terms of this wave vector, for every such pure ( i.e. not given by an electromagnetic wave ) gravitational wave orbit  but how?  ,curvature' means curvature structure
in the following 
Classification Still Unsolved  But which Weyl curvatures are determined by wave vectors? Since the (full) Weyl group acts transitively on the nullspace, and group actions permute with Weyl and Einstein projectors in Singer & Thorpe's curvature space, one perhaps can show, that there is only one wave orbit in the subspace of Weyl curvatures. But it is unlikely that one such orbit exhausts the space of Weyl curvatures, since this is a ( awfully high ) n(n+1)(n+2)(n3)/12  dimensional subspace in the whole curvature space, for the dimension of which one has to add n(n+1)/2, the dimension of the ( Jordan algebra of ) <,>selfadjoint endomorphisms. If there is only one (gravitational) wave orbit there arises the question, what are the other orbits? Are one or two ( the number depends on the signature of the underlying bilinear form ) given by timelike or spacelike vectors of the underlying vector space? For this we need an explicite construction of a curvature in terms of these vectors, like we have one in the three cases by vectors and endomorphisms above. There is such a construction by using tensor products of such a vector, to get an endomorphism and insert it into these three constructions. In the 'mixed' case  the electromagnetic one  we cannot arrive by this construction in the space of Weyl curvatures. In the other cases we have to find 'odd' constructions in order to arrive only in the space of Weyl curvatures.  is there a further decomposition of the space of Weyl curvatures ? 
Categories Matter!  Electromagnetic orbits are given by two physical tensor fields, the field strengths E ( a 3vector E together with a skew 3×3 matrix H ) and the intensities D ( a 3vector D together with a skew 3×3 matrix B ) in media without structure and special relativity's flat Minkowski space with a linear dependence in between. Physicists take them as skew 4×4 matrices – which is erronous! Because this means the introduction of a positivedefinite metric in spacetime and an Äther(category) which – doesn't exist! We have to formulate electrodynamics entirely in the Minkowski category, which means that E and B have to be <,>skewadjoint! Even exterior algebra formulations by exterior differential forms lead into a dead end – we have to take a Clifford algebra ( over the Minkowski space ) instead, wellknown in physics from Fermi statistics.  to get rid of Äther, reformulate Maxwell's equations 
Petrov's Classification  Petrov's classification by bivectors is one in the n(n+1)/2dimensional Jordan algebra of <,>selfadjoint endomorphisms, since those are generated linearly by pairs of vectors of VV.It is mapped [ T84 ] by the linear map Ω onto the cosmological subspace of the curvature space. Since this Jordan algebra is one of the classical real simple ones, this classification is given by Dynkin diagrams. However, the complementary subspace of the Weyl curvature structures is not arrived at by such an easy map. Clearly Petrov's eigenvaluebased curvature classification is one by orbits of the Weyl group.  has to be generalized to the whole curvature space 
Embedding Electromagnetic Fields into Gravity  Implementing an electromagnetic field into general relativity usually is done by inserting its energy momentum tensor into Einsteins gravitational field equations as a source ( i.e. on the right hand side ) and solving for metric, connection and curvature. But this only is an indirect method. Hence there is the question whether there exists a direct method, starting from a <,>skewadjoint electromagnetic field strength E and an intensity D? So far we don't know any such direct construction. If this energy momentum tensor is that one of an electromagnetic field only, like in most regions of the universe, we have the electrovac special case Lichnérowics [ Lic ] has given an axiomatic characterization of electromagnetic waves, like that one above of graviational waves: The pair (E, D) is said to be an electromagnetic wave if there exists a (necessarily) lightlike vector l such that for any x,y,z ∈ VVand the same equations for the intensity D. Clearly symmetrization is equivalent toCertainly for most physical media D = ε E and especially for the vacuum physisists write εo for this constant. That it is not an universal constant, but depends on the curvature at any position in spacetime follows from  must have a better solution
el.mag. field ⊑ grav.field ⊔ ⊔ el.mag. wave ⊑ grav.wave 
Post's Gravity Contribution to Electromagnetic Intensities  [ Post p 26] remarks that if E and D are linearly connected ( like in most media on earth ) the 4tensor relating them has the same symmetries as a curvature tensor, i.e. fulfills Singer & Thorpe's curvature axioms. Since he starts from only one field strength E, his method leads to a first question: Start in between Singer & Thorpe's and Nomizu's curvature constructions by one <,>selfadjoint A ( Post has the identity ) and a <,>skewadjoint electromagnetic field E, which, when inserted into the given curvature, delivers the <,>skewadjoint intensity D. In [ T84 p 26] it was shown that for this ,mixed' case, there are only solutions in the space of nonWeyl curvatures, none in the space of Weyl ones, and if A especially is the identity, even these two vanish. In fact, the only possible curvatures constructed from such A's and E's are of the form E•A and the trace thereof, where the • is the natural ( Lie on Jordan ) algebraical coadjoint adaction by inner derivations [T84]. So the only nontrivial embedding of electromagnetics into gravity remains Nomizu's! But this only is the case because all three constructions, that by two <,>selfadjoint or that by two −skewadjoint endomorphisms or the mixed one, start from the trivial 1dimensional scalar curvature  more general ones do not work.
Post's construction inverted leads to more: Given an electromagnetic field strength E in a gravitational field, given by its curvature R, construct the electromagnetic field intensity B in the same way as the three mathematical approaches from the trivial curvature, but this time more general from the given curvature R. Clearly this modification of R by E no longer is a curvature: Two of Singer & Thorpe's curvature axioms are not fulfilled – <,>skew symmetry ( because E is <,>skewadjoint B(x,y) can't be ) and the Bianchiidentity ( that's why they are not interesting for mathematics  they contribute nothing to the structure theory ). However, it is clear from the construction that the third still holds, i.e. the outcome B is in the pseudoorthogonal Lie algebra and hence may be interpreted as an electromagnetic field intensity. This has a physical consequence: The full gravitational field ( not only its scalar part, i.e. the trivial curvature ) contributes to electromagnetic field intensities. Since there is no point in spacetime without gravitation this contribution never vanishes. Half way between two adjacent galaxies, this contribution is neglectible, but near the center of a galaxy, near the event horizon of a black hole, for a strong gravitational wave or in the first moments of a big bang model these curvaturemodified intensities B can have measurable physical effects. Only in a universe with constant curvature, actually that one by Irving Segal, studied on this webpage, this contribution reduces to a ( in some positions of spacetime huge ) number, the curvature radius of that position.  B(x,y) = R(x,Ey)
for any vectors x, y of Minkowski space can have drastic physical effects in regions of strong gravitation 
Cosmological Models  Physical spacetime is not a vector space as in this study sofar, but a pseudoRiemannian manifold of dimension 4 and signature +,−,−,−. <,> becomes the eigentime, vector fields, i.e. sections of the tangent bundle take the role of the elements of the vector space VVabove, over which there are bundles whose fibres are curvature spaces. Curvature structures are sections in such a curvature bundle. A dynamical system is a geodesic ( with respect to <,> ) in spacetime, giving rise to a LeviCivita connection, which in turn gives rise to a curvature structure. GGis the structure group of these bundles. A cosmological model is a spacetime, whose curvature sections lie in one GG  orbit.Hence the classification of these orbits in the curvature space is exactly the classification of cosmological models. Needless to say, actual spacetime is not a cosmological model, but in each point of spacetime there is a cosmological model, which approximates actual spacetime better than the flat tangent Minkowski space. Equivalently one can develop the concept of local curvature structure, as Loos has given the concept of locally symmetric spaces in his books.  can actual spacetime even be better approximated by using third order derivatives ? 
 Also Possible
quantization  Assuming a big bang cosmology, in the first moments of the universe strong electromagnetic fields alone can have led to riddles in the cosmic back ground radiation.  
