In this worksheet we use the 15-dimensional real Lie algebra su(2, 2) to illustrate some important points regarding the general structure theory and classification of real semi-simple Lie algebras.
1. Recall that a real semi-simple Lie algebra g is called a compact Lie algebra if the Killing form is negative definite. The Lie algebra g is compact if and only if all the root vectors for any Cartan subalgebra are purely imaginary. However, if the root vectors are purely imaginary for some choice of Cartan subalgebra it is not necessarily true that the Lie algebra is compact.
2. A real semi-simple Lie algebra g is called a split Lie algebra if there exists a Cartan subalgebra such that the root vectors are all real. However, it is not true that if the root vectors are all real with respect one choice of Cartan subalgebra then they are real with respect to any other choice.
3. Points 1 and 2 reflect the fact that, for complex Lie algebras, all Cartan subalgebras are equivalent in the sense that they may be mapped into each other by a Lie algebra automorphism. This is not true for real semi-simple Lie algebras.
4. To properly analyze the structure of a given real Lie algebra 𝔤 , one must first calculate a Cartan decompositiong= t + p. The Killing form of g is negative-definite on t and positive-definite on p. Next one must chose a Cartan subalgebra h which [i] is aligned with the Cartan decomposition and [ii] such that the non-compact part of h ∩ p is of maximal dimension. Such Cartan subalgebras are said to be maximally non-compact. With such a Cartan subalgebra one obtains the minimum number of pure imaginary (or compact) roots, one can draw the Satake diagram and thereby obtain the correct real classification of the Lie algebra.
We illustrate these points with the real Lie algebra su(2, 2) .
Anderson, Ian M., "Cartan Subalgebras, Compact Roots and the Satake Diagram for su(2, 2)" (2014). Tutorials on... in 1 hour or less. Paper 2.