For compact Lie groups, is $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$ equivalent to being semisimple?

Question

In Ana Cannas da Silva’s Lectures on Symplectic Geomertry, page 167, semisimplicity is defined in the restricted setting of compact Lie groups by

A compact Lie group $G$ is semisimple if $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$.

Is this definition consistent with the “normal” definition of semisimplicity, and if so, does someone have a reference for it?

Answer 1

Yes, a connected compact Lie group is the product of a semi-simple Lie group and a torus $T^n$, thus its Lies algebra $g$ is the sum of a semi-simple Lie algebra $s$ and a commutative algebra $c$, such that $[s,c]=0$, thus $[s+c,s+c]=s$.

https://en.wikipedia.org/wiki/Compact_Lie_algebra#Definition

Answer 2

A connected compact Lie group $G$ has a reductive Lie algebra $L=[L,L]\oplus Z(L)$, where $[L,L]$ is semisimple. Furthermore it has finite center $Z(G)$ if and only if $[G,G]=G$, i.e., if and only if $[L,L]=L$. It follows that a connected compact Lie group $G$ satisfies $[L,L]=L$ if and only if $Z(L)=0$, so that $L=[L,L]$ is semisimple, or equivalently $G$ is semisimple.