The complex Lie algebra can be represented as the space of complex linear combinations of the three matrices

which satisfy the well known commutation relations A crucial property of these commutation relations is that the structure constants defined by the brackets are all real. Thus by restricting the scalars in the linear combinations from the complex to the real numbers, we still obtain closure for the Lie bracket on real combinations of and , defining thereby a real form of the complex Lie algebra : the real Lie algebraAnother choice of generators that, similarly, leads to a real Lie algebra consists in taking times the Pauli matrices , , , i.e.,

The real linear combinations of these matrices form the familiar Lie algebra (a real Lie algebra, even if some of the matrices using to represent it are complex). This real Lie algebra is non-isomorphic (as a real algebra) to as there is no real change of basis that maps on a basis with the commutation relations. Of course, the two algebras are isomorphic over the complex numbers.

Let be a subalgebra of . We say that is a Cartan subalgebra of if it is a Cartan subalgebra of when the real numbers are replaced by the complex numbers. Two Cartan subalgebras and of are said to be equivalent (as Cartan subalgebras of ) if there is an automorphism of such that .

The subspace constitutes clearly a Cartan subalgebra of . The adjoint action of is diagonal in the basis and can be represented by the matrix

Another Cartan subalgebra of is given by , whose adjoint action with respect to the same basis is represented by the matrix Contrary to the matrix representing , in addition to 0 this matrix has two imaginary eigenvalues: . Thus, there can be no automorphism of such that , since has the same eigenvalues as , implying that the eigenvalues of are necessarily real ().Consequently, even though they are equivalent over the complex numbers since there is an automorphism in that connects the complex Cartan subalgebras and , we obtain

The real Cartan subalgebras generated by and are non-isomorphic over the real numbers.

The Killing form of reads explicitly

in the basis . The Cartan subalgebra is spacelike while the Cartan subalgebra is timelike. This is another way to see that these are inequivalent since the automorphisms of preserve the Killing form. The group of automorphisms of is , while the subgroup of inner automorphisms is the connected component of . All spacelike directions are equivalent, as are all timelike directions, which shows that all the Cartan subalgebras of can be obtained by acting on these two inequivalent particular ones by , i.e., the adjoint action of the group . The lightlike directions do not define Cartan subalgebras because the adjoint action of a lighlike vector is non-diagonalizable. In particular and are not Cartan subalgebras even though they are Abelian.By exponentiation of the generators and , we obtain two subgroups, denoted and :

The subgroup defined by Equation (6.14) is noncompact, the one defined by Equation (6.15) is compact; consequently the generator is also said to be noncompact while is called compact.

The Killing metric on the group is negative definite. In the basis , it reads

The corresponding group obtained by exponentiation is , which is isomorphic to the 3-sphere and which is accordingly compact. All directions in are equivalent and hence, all Cartan subalgebras are conjugate to . Any generator provides by exponentiation a group isomorphic to and is thus compact.

Accordingly, while admits both compact and noncompact Cartan subalgebras, the Cartan subalgebras of are all compact. The real algebra is called the compact real form of . One often denotes the real forms by their signature. Adopting Cartan’s notation for , one has and . We shall verify before that there are no other real forms of .

Within , one may express the basis vectors of one of the real subalgebras or in terms of those of the other. We obtain, using the notations and :

Let us remark that, in terms of the generators of , the noncompact generators and of are purely imaginary but the compact one is real. More precisely, if denotes the
conjugation^{20}
of that fixes , we obtain:

The two conjugations and of associated with the real subalgebras and of commute with each other. Each of them, trivially, fixes pointwise the algebra defining it and globally the other algebra, where it constitutes an involutive automorphism (“involution”).

The Killing form is neither positive definite nor negative definite on : The symmetric matrices have positive norm squared, while the antisymmetric ones have negative norm squared. Thus, by changing the relative sign of the contributions associated with symmetric and antisymmetric matrices, one can obtain a bilinear form which is definite. Explicitly, the involution of defined by has the feature that

is positive definite. An involution of a real Lie algebra with that property is called a “Cartan involution” (see Section 6.4.3 for the general definition).The Cartan involution is just the restriction to of the conjugation associated with the compact real form since for real matrices . One says for that reason that the compact real form and the noncompact real form are “aligned”.

Using the Cartan involution , one can split as the direct sum

where is the subspace of antisymmetric matrices corresponding to the eigenvalue of the Cartan involution while is the subspace of symmetric matrices corresponding to the eigenvalue . These are also eigenspaces of and given explicitly by and . One has i.e., the real form is obtained from the compact form by inserting an “” in front of the generators in .

Let us close these preliminaries with some remarks.

- The conjugation allows to define a Hermitian form on :
- Any element of the group can be written as a product of elements belonging to the subgroups , and (Iwasawa decomposition),
- Any element of is conjugated via to a multiple of , so, denoting by the (maximal) noncompact Cartan subalgebra of , we obtain
- Any element of can be written as the product of an element of and an element of
. Thus, as a consequence of the previous remark, we have
(Cartan)
^{21}. - When the Cartan subalgebra of is chosen to be , the root vectors are and . We obtain the compact element , generating a non-equivalent Cartan subalgebra by taking the combination Similarly, the normalized root vectors associated with are (up to a complex phase) : Note that both the real and imaginary components of are noncompact. They allow to obtain the noncompact Cartan generators by taking the combinations

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