Adjoint representation in second tensor power of faithful representation Let $ G $ be a semisimple Lie group and $ (\pi,V) $ a faithful finite dimensional representation of $ G $. Consider the action of $ G $ on $ V \otimes V $ by
$$
g \cdot (v_1 \otimes v_2)= gv_1 \otimes gv_2
$$
Is it always true that one of the irreducible sub representations of $ V \otimes V $ is the adjoint representation of $ G $?
Thoughts so far:
This is true for $ G=SU_n $ and the natural $ n $ dimensional faithful representation. Indeed in this case $ V \otimes V $ decomposes as a direct sum of the adjoint representation (dimension $ n^2-1 $) with one copy of the trivial representation (dimension 1).
 A: The claim is false in general; see below for some counterexamples for simple $G$. As mentioned in the comments, by definition any faithful representation $V$ of $\mathfrak{g}$ defines an embedding $\mathfrak{g} \hookrightarrow \operatorname{End} V \cong V \otimes V^*$ of the adjoint representation, $\mathfrak{g}$, that is, $V \otimes V^*$ always contains an isomorphic copy of the adjoint representation.
In the case that $V$ admits a nondegenerate $G$-invariant bilinear form $\beta$, that form defines an isomorphism of $G$-representations $V \to V^*$ by $v \mapsto \beta(v, \,\cdot\,)$, i.e.,"raising an index", and so in that case $V \otimes V \cong V \otimes V^*$, and in particular $V \otimes V$ contains a copy of $\mathfrak{g}$. This condition applies, for example, to all faithful representations of simple Lie groups of type $B, C, F$, and $G$, as well as $E_7$ and $E_8$.
We can find simple counterexamples to the original claim by looking for simple Lie algebras with outer automorphisms satisfying a particular condition.

*

*For $\operatorname{SL}_n(\Bbb C)$, $n > 2$, and its standard standard representation $V$, the tensor square decomposes into irreducibles as $$V \otimes V \cong \operatorname{Sym}^2 V \oplus \bigwedge\!{}^2 \,V ,$$ neither summand of which is the adjoint representation. (The decomposition still holds for $n = 2$, but in that case the adjoint representation is $\mathfrak{sl}_2(\mathbb C) \cong \operatorname{Sym}^2 V$.)


*For $G = \operatorname{SO}_{4 m + 2}(\mathbb C)$, $m > 0$, and $V$ and $S_\pm$ the standard and positive (negative) spin representation, respectively, the tensor square of $S_\pm$ decomposes into irreducibles as $$S_\pm \otimes S_\pm \cong V \oplus \bigwedge\!^3 V \oplus \cdots \oplus \bigwedge\!^{2 m - 3} V \oplus \bigwedge\!^{2 m - 1} V \oplus \bigwedge\!^{2 m + 1}_\pm V,$$ where $\bigwedge\!^{2m + 1}_\pm V$ is the representation of (anti-)self-dual $(2m + 1)$-forms, but the adjoint representation is $\mathfrak{so}_{4 m + 2}(\mathbb C) \cong \bigwedge\!^2 V$ .


*For a more exotic example, take $G$ to be the exceptional complex simple Lie group $\operatorname{E}_6$, and let $V$ denote either of the $27$-dimensional fundamental representations. Then, the tensor square of $V$ decomposes into irreducibles as:
$$V \otimes V = W \oplus V^* \oplus \operatorname{Sym}^2 V ,$$ where $W$ is the (only) $351$-dimensional subrepresentation of $\bigwedge^2 V$. (The representation $W$ is itself also a counterexample, but the decomposition of $W \otimes W$ into irreducibles is more complicated.)
