S-subalgebra and R-subalgebra.

Question

Given the characteristics of a Cartan subalgebra H of a semisimple Lie algebra; H is abelian, it stabilises the root spaces generated with respect to it. It appears to me that all subalgebras of a complex semisimple Lie algebra are regular; since any subalgebra is a subspace containing some root spaces and some elements of H, it will therefore be normalised by H. What then is the idea of S-subalgebras; subalgebras not contained in a proper regular subalgebra?

Answer 1

No, not all subalgebras of complex semisimple Lie algebras are regular - see here for an example (which is the non-regular subalgebra $A_1\oplus A_2$ in $A_5$). Here $A_n$ denotes the simple Lie algebra $\mathfrak{sl}_{n+1}(K)$.