Lie algebra structure on $\mathbb{C}[G]$

Question

Let $G$ be a finite group. The group algebra $\mathbb{C}[G]$ has a Lie algebra structure inherited by its associative algebra structure, which is given by $$[g,h]=gh-hg.$$ It is not too hard to check that the center $Z(\mathbb{C}[G])$ consists of the subspace of functions constant on conjugacy classes (upon identification of $\mathbb{C}[G]$ with the space of functions $G \to \mathbb{C}$). Moreover, $[\mathbb{C}[G],\mathbb{C}[G]]$ is the subspace of functions whose sum on every conjugacy class is zero. In particular we have $\mathbb{C}[G]=Z(\mathbb{C}[G])\oplus [\mathbb{C}[G],\mathbb{C}[G]]$, so it is reductive.

Question: What can we say about the structure of the semi-simple part $[\mathbb{C}[G],\mathbb{C}[G]]$, in particular what is its corresponding Dynkin diagram?

See also this MO-post, where in the first case Lie algebra is a sum of general linear Lie algebras. Of course then the semisimple part is $\bigoplus_V \mathfrak{sl}(V)$. See also this linked paper. — Dietrich Burde, Apr 23 '22 at 16:14
Although it is true in hindsight as per Dietrich Burde's link (structure of semisimple associative algebras), I do not think you can conclude that the Lie algebra is reductive from just what you write. — Torsten Schoeneberg, Apr 23 '22 at 17:34
Isn't it true that reductive is equivalent to $\mathfrak{g} = [\mathfrak{g},\mathfrak{g}]\oplus Z(\mathfrak{g})$? — Antoine Labelle, Apr 23 '22 at 17:39
I think both answers in https://math.stackexchange.com/q/2424899/96384 show that is not the case (they have trivial centre and are perfect but not semisimple). — Torsten Schoeneberg, Apr 24 '22 at 05:54

Lie algebra structure on $\mathbb{C}[G]$

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