Let g be a split, semi-simple real Lie algebra. This implies, in particular, that there exists a Cartan subalgebra for g for which the associated root space decomposition of g is real. For such Lie algebras there is a very special basis called the Chevalley basis. In this worksheet we calculate the Chevalley basis for the Lie algebra sp(6, R) and illustrate the various properties of this basis.
Anderson, Ian M., "The Chevalley Basis for a Split Semi-simple Lie Algebra" (2015). Tutorials on... in 1 hour or less. Paper 7.