Lie groups and Lie algebras: sl(2,C) subalgebras in general, 1

We begin the construction of sl(2,C) subalgebras in the Lie algebra of a compact semisimple group.

[@khoanguyen5321]

I have heard that SL(2,C), depending on chosen of generators to be purely real or imaginary, we will get SU(2) (compact) and SL(2,R) (noncompact) accordingly. Is that correct? How can I show it>