Adjoints in symmetric squares of Lie algebra representations

Caveat: we learned post-factum that most of these results are not novel. We are keeping this paper for continuity reasons. Given finite-dimensional complex representations $V$ and $V^{\mathrm{\prime }}$ of a simply-connected semisimple compact Lie group $G$, we determine the dimension of the $G$-invariant subspace of $\mathrm{a}\mathrm{d}\mathrm{j}(G)\otimes V\otimes V^{\mathrm{\prime }}$, of $\mathrm{a}\mathrm{d}\mathrm{j}(G)\otimes S^{2}V$, and of $\mathrm{a}\mathrm{d}\mathrm{j}(G)\otimes {\mathrm{\Lambda }}^{2}V$, where $\mathrm{a}\mathrm{d}\mathrm{j}(G)$ is the adjoint representation. In other words we derive the multiplicity with which summands of $\mathrm{a}\mathrm{d}\mathrm{j}(G)$ appear in a tensor product $V\otimes V^{\mathrm{\prime }}$ or (anti)symmetric square $S^{2}V$ or ${\mathrm{\Lambda }}^{2}V$. We find in particular that the dimension of the $G$-invariant subspace of $\mathrm{a}\mathrm{d}\mathrm{j}(G)\otimes S^{2}V$ is larger than (resp. smaller or equal to) that of $\mathrm{a}\mathrm{d}\mathrm{j}(G)\otimes {\mathrm{\Lambda }}^{2}V$ for a symplectic (resp. orthogonal) representation $V$.