General Outline
Assume one has a decomposition of the tensor product of the adjoint representation into irreducible components:
$\mathfrak{g} \otimes \mathfrak{g} = \bigoplus_{\lambda} V_{\lambda}^{\oplus m(\lambda)} $
where $\mathfrak{g}$ is a semi-simple Lie algebra, $\lambda$ denotes the highest weight of the irreducible representation in $V_{\lambda}$ and $m(\lambda)$ is the multiplicity of the irrep $\lambda$ in the decomposition.
We can also decompose $\mathfrak{g} \otimes \mathfrak{g}$ into it's symmetric square and second exterior power (see ref 1)
$\mathfrak{g} \otimes \mathfrak{g} = Sym^2(\mathfrak{g}) \oplus \Lambda^2(\mathfrak{g})$.
I am intersted in computing $Sym^2(\mathfrak{g})$.
Question I
Is there a (computationally efficient) way of checking which irreps $V_{\lambda}$ occurring in the decomposition of $\mathfrak{g} \otimes \mathfrak{g}$ contribute to the symmetric square $Sym^2(\mathfrak{g})$?
Refined Question II
I am particularly interested in the case of $\mathfrak{g} = \mathfrak{su}(n)$. Any generalities applying beyond this case would be welcome.
Attempt At a Solution
The $n$th symmetric tensor power $Sym^n(V_{\lambda})$ of a irrep $V_{\lambda}$ is defined as the subrepresentation of $V_{\lambda}^{\otimes n}$ which is invariant under the permutation group on $n$ elements, $S_n$. For the case at hand, we have $V_{\lambda} = \mathfrak{g}$ and $n=2$ where $S_2$ has two elements being the identity and an order two element $\sigma$ with eigenvalues $\pm 1$. The $+1$ eigenspace is the subrepresentation $Sym^2 (\mathfrak{g})$ while the $-1$ eigenspace is that of the exterior power $\Lambda^2(\mathfrak{g})$. For $n=2$ these are orthogonal subspaces so we have the decomposition given above: $$\mathfrak{g} \otimes \mathfrak{g} \cong Sym^2(\mathfrak{g}) \oplus \Lambda^2(\mathfrak{g}).$$ If this approach is correct, then the answer to I can be given by computing which irreps occurring in the decomposition $\bigoplus_{\lambda} V_{\lambda}^{\oplus m(\lambda)}$ are invariant under $\sigma$. Such irreps are by definition contained in $Sym^2(\mathfrak{g})$.
Resulting Question III
If this approach is correct, my question is then how to actually check which irreps contained in $\bigoplus_{\lambda} V_{\lambda}^{\oplus m(\lambda)}$ are invariant under $\sigma$ in terms of the irrep data. That is, I am wondering if this can be done by simply knowing the highest weight $\lambda$ for $V_{\lambda}$ or if I need to write down the tensor structure of $V_{\lambda}$ in terms of projections onto $\mathfrak{g} \otimes \mathfrak{g}$ elements. I'd prefer an "abstract" approach which only requires knowledge of something like the highest weights $\lambda$ occurring in the decomposition since writing down projectors and deriving the explicit tensor structure can be difficult and tedious. I feel one should also be able to check this simply by knowing the Young tableaux for $V_{\lambda}$, but I am not seeing how to achieve this.
Thank you for any feedback.