Let be a positive integer. For a non-degenerate symmetric matrix , define called the orthogonal Lie algebra with respect to . The standard definition of the special orthogonal Lie algebra is given by where “special” implicitly indicates the fact that .
There is another common definition of the special orthogonal Lie algebra, which we refer to here as the twisted special orthogonal Lie algebra, defined as the Lie algebra The following theorem shows that these two definitions are equivalent.
TheoremLet be a positive integer. Let be two non-degenerate symmetric matrices. Then there exists an invertible matrix such that . Under this transformation, is an isomorphism of Lie algebras.
The existence of in this theorem follows from the fact that any two non-degenerate symmetric matrices are congruent over .
The standard definition has the benefit of being simpler, and its simplicity (for ) is easier to derive (by brute force calculation). However, the Cartan subalgebra and its Cartan decomposition are very difficult to describe:
ExerciseLet be a positive integer. Prove that is a Cartan subalgebra of , where .
The twisted definition has the advantage that the diagonal matrices in form its Cartan subalgebra , and the standard basis of coincides with its weights with respect to .