Irreducibility of the $\mathfrak{g}$-module $\mathfrak{o}(k)/ad(\mathfrak{g})$

Question

For a simple lie algebra $\mathfrak{g}$ over a field of characteristic 0, define $\mathfrak{o}(k)$ to be the orthogonal lie algebra with respect to the Killing form.

In the proof of Theorem 2 in the following paper, https://arxiv.org/pdf/math/0407240.pdf the author mentions that the following is true,

The $\mathfrak{g}$ module $\mathfrak{o}(k)/ad(\mathfrak{g})$ is irreducible if $\mathfrak{g}$ is not of type A while it is a direct sum $W \oplus W^∗ $for some non-self-dual module $W$ if $\mathfrak{g}$ is of type A.

I am unable to see why this is true. Can someone furnish a proof or point to references?

I've also asked it here - https://math.stackexchange.com/questions/3335347/irreducibility-of-the-mathfrakg-module-mathfrakok-ad-mathfrakg

Answer 1

Note that ${\mathfrak o}(k)\cong \wedge^2 {\mathfrak g}$. It has ${\mathfrak g}$ as a summand, coming from the Lie bracket $\wedge^2 {\mathfrak g} \rightarrow {\mathfrak g}$ . Calculation of the rest is an easy case-by-case exercise. For instance, it is done by Reeder in https://mathscinet.ams.org/mathscinet-getitem?mr=1437204