Given any group $ G $ of $ n \times n $ complex matrices there is a natural $ n $ dimensional representation of $ G $ on $ V=\mathbb{C}^n $. In this case the representation
$$
V \otimes V^* \cong \mathfrak{gl}(V)
$$
can be naturally viewed as $ G $ acting by conjugation on the $ n^2 $ dimensional space of all $ n \times n $ complex matrices. Whenever $ G $ is compact then one the direct summands in the decomposition of $ V \otimes V^* \cong \mathfrak{gl}(V) $ will always be the complexification of the adjoint representation of $ G $.
For example, consider the natural module $ (SU_n, V=\mathbb{C}^n) $. Then $ V \otimes V^* \cong \mathfrak{gl}(V) $ decomposes as the $ n^2-1 $ dimensional adjoint irrep $ \mathfrak{sl}_n(\mathbb{C}) $ (which is the complexification of $ \mathfrak{su}_n $) together with a 1 dimensional trivial irrep (the span of the identity matrix $ \sum_{i=1}^n e_i \otimes e_i^* $).
For the natural module $ (SO_n, V=\mathbb{C}^n) $ I'm curious about the direct sum decomposition of $ V \otimes V^* \cong \mathfrak{gl}(V) $ since it is $ n^2 $ dimensional it must include a lot of other interesting stuff beyond the $ \frac{n(n-1)}{2} $ dimensional adjoint representation.
Similarly, for the natural module $ (Sp_n, V=\mathbb{C}^{2n}) $ I'm curious about the direct sum decomposition of $ V \otimes V^* \cong \mathfrak{gl}(V) $ since it is $ 4n^2 $ dimensional it must include a lot of other interesting stuff beyond the $ n(2n+1) $ dimensional adjoint representation.
I'm sure all this is well known, so I would also be happy to accept a reference.
Best Answer
A standard reference for these results is Fulton & Harris's Representation Theory, $\S\S$16–20.
In both the orthogonal and symplectic cases, dualizing using the (respectively, symmetric and asymmetric) bilinear form preserved by the group identifies $\mathfrak{gl}(V) \cong V \otimes V^*$ with $V \otimes V$, which hence decomposes as $$\mathfrak{gl}(V) \cong \operatorname{Sym}^2 V \oplus \bigwedge\!^2\, V .$$
In the orthogonal case, $n \geq 3$, $\bigwedge\!^2 \,V$ is isomorphic to the adjoint representation, $\mathfrak{so}_n$, and $\operatorname{Sym}^2 V$ decomposes as the direct sum of (1) the line $\Bbb C g$ of scalar multiples of the symmetric bilinear form $g$ preserved by $SO_n$ and (2) the hyperplane $\operatorname{Sym}^2_\circ V$ of symmetric $2$-tensors tracefree with respect to $g$, i.e., those satisfying $g_{ab} T^{ab} = 0$. In summary, $$\mathfrak{gl}_n = \mathfrak{so}_n \oplus \operatorname{Sym}^2_\circ V \oplus \Bbb C .$$
For $n \neq 4$ the three summands we have identified are all irreducible.
For $n = 4$, $g$ and the orientation defined by $SO_4$ on $V$ together define the Hodge star endomorphism, $\ast: \bigwedge\!^2 \,V \to \bigwedge\!^2 \,V$, which satisfies $\ast^2 = \operatorname{id}$, and so $\bigwedge\!^2 \,V$ decomposes further as a direct sum of the $(\pm 1)$-eigenspaces $\bigwedge\!^2_\pm \,V$ of $\ast$, both of which are irreducible and which have dimension $3$. (That the adjoint representation is not irreducible in this case is a consequence of the fact that $SO_4$ is not simple; indeed $\mathfrak{so}_4 \cong \mathfrak{so}_3 \oplus \mathfrak{so}_3$.) We have: $$\mathfrak{gl}_4 = \overbrace{\mathfrak{so}_3 \oplus \mathfrak{so}_3}^{\mathfrak{so}_4} \oplus \operatorname{Sym}^2_\circ V \oplus \Bbb C .$$
For $n = 3$, the three summands are again irreducible, but notice that in this case the Hodge star operation defines an isomorphism between $\mathfrak{so}_3 \cong \bigwedge^2 V$ and $V$.
In the symplectic case the reverse is true: $\operatorname{Sym}^2 V$ is isomorphic to the adjoint representation, $\mathfrak{sp}_n$, and $\bigwedge\!^2\,V$ decomposes as the direct sum of (1) the line $\Bbb C \omega$ of scalar multiples of the symplectic form $\omega$ preserved by $Sp_n$ and (2) the hyperplane $\bigwedge\!^2{\!\!}_\circ\,V$ of skew $2$-tensors tracefree with respect to $\omega$, i.e., those satisfying $\omega_{ab} T^{ab} = 0$. So, $$\mathfrak{gl}_n \cong \mathfrak{sp}_n \oplus \bigwedge\!^2{\!\!}_\circ\,V \oplus \Bbb C .$$
Notice that for $n = 1$ all skew $2$-tensors are multiples of the dual of the symplectic form, leaving $$\mathfrak{gl}_2 \cong \mathfrak{sp}_1 \oplus \Bbb C .$$ In all cases the summands are irreducible.
Incidentally, one can carry out decompositions of semisimple representations into irreducible submodules using Lie or SAGE, both free software.