6.3 The compact and split real forms of a semi-simple Lie algebra

We shall consider here only semi-simple Lie algebras. Over the complex numbers, Cartan subalgebras are “unique”. These subalgebras may be defined as maximal Abelian subalgebras such that the transformations in are simultaneously diagonalizable (over ). Diagonalizability is an essential ingredient in the definition. There might indeed exist Abelian subalgebras of dimension higher than the rank (= dimension of Cartan subalgebras), but these would involve non-diagonalizable elements and would not be Cartan subalgebras.

We denote the set of nonzero roots as . One may complete the Chevalley generators into a full basis, the so-called Cartan–Weyl basis, such that the following commutation relations hold:

where is defined by duality thanks to the Killing form , which is non-singular on semi-simple Lie algebras:
and the generators are normalized according to (see Equation (6.43))
The generators associated with the roots (where need not be a simple root) may be chosen such that the structure constants satisfy the relations
where the scalar product between roots is defined as
The non-negative integers and are such that the string of all vectors belongs to for ; they also satisfy the equation . A standard result states that for semi-simple Lie algebras
from which we notice that the roots are real when evaluated on an -generator,

An important consequence of this discussion is that in Equation (6.32), the structure constants of the commutations relations may all be chosen real. Thus, if we restrict ourselves to real scalars we obtain a real Lie algebra , which is called the split real form because it contains the maximal number of noncompact generators. This real form of reads explicitly

The signature of the Killing form on (which is real) is easily computed. First, it is positive definite on the real linear span of the generators. Indeed,
Second, the invariance of the Killing form fixes the normalization of the generators to one,
since
Moreover, one has if . Indeed maps into , i.e., in matrix terms has zero elements on the diagonal when . Hence, the vectors are spacelike and orthogonal to the vectors , which are timelike. This implies that the signature of the Killing form is
The split real form of is “unique”.

On the other hand, it is not difficult to check that the linear span

also defines a real Lie algebra. An important property of this real form is that the Killing form is negative definite on it. Its signature is
This is an immediate consequence of the previous discussion and of the way is constructed. Hence, this real Lie algebra is compact. For this reason, is called the “compact real form” of . It is also “unique”.