New posts in Semisimple Lie Algebras

How are $SU(n)$, $SL(n)$ and $\mathfrak{sl}(n,\mathbb{C})$ related?

Finite dimensional irreducible representations of a semisimple Lie Algebra separate points of the universal enveloping algebra.

If $\mathfrak{g}=\bigoplus\mathfrak{g}_i$ is a semisimple Lie algebra, why does $\mathfrak{h}=\bigoplus\left(\mathfrak{h}\cap\mathfrak{g}_i\right)$?

Is every element of a complex semisimple Lie algebra a commutator?