For a simple lie algebra  $\mathfrak{g}$, 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