A classical problem of invariant theory is to describe the space $((V^*)^{\otimes r} \otimes V^{\otimes s})^G$ in terms of generators and relations, or equivalently to determine generators and relations of $\text{End}_GV^{\otimes r}$, for $V$ some $K$-dimensional vector space (often $K = \mathbb{C}$). The classical groups $GL(V)$, $O(V)$, and $Sp(V)$ have "fundamental theorems" that describe the above mentioned spaces, proven via symmetric group representations or via Brauer diagrams and functorial correspondences. An example formulation of the first fundamental theorem I am working with is
The unique covariant functor $F$ from the Brauer category (with respect to a certain fixed element of $\mathbb{C}$) to the subcategory of $G$-modules $V^{\otimes r}$ is full, i.e. induced maps from Brauer algebras $B_r^s \to \text{Hom}_G(V^{\otimes r}, V^{\otimes s})$ are surjective for all $r,s$.
There are many proofs for this statement (and its equivalent formulations, e.g. via matrices giving a generating set of $((V^{\otimes 2r})^*)^G$), one of which uses the following idea/lemma:
Let $E^+$ be the subset of $\text{End}(V)$ which consist of self-adjoint operators (for $f \in \text{End}(V)$, its adjoint is the unique $f^\alpha$ for which $(fv,w) = (v,f^\alpha w)$ for all $v,w \in V$ - obviously this definition depends on if the form on $V$ is symmetric or skew-symmetric). Let $\omega: \text{End}(V) \to E^+$ be the map given by sending $f$ to $f^\alpha f$. Let $\phi : \text{End}(V) \to \mathbb{C}$ be such that $\phi(gf) = \phi(f)$ for all $g \in G$. Then there exists some $\psi \in \mathbb{C}[E^+]$ for which $\phi(f) = \psi(\omega(f))$ for all $f \in E$.
My notes say this lemma is due to an idea of Atiyah's, possibly appearing in a paper of Atiyah and two others (initials ABP - possibly Bott and Patodi?), but I cannot find exactly where, nor a proof of it.
This is in fact suggested in "On the heat equation and the Index Theorem" by Atiyah, Bott, and Patodi (1973) in Inventiones Mathematicae, Vol. 19 pp 279-230.
The result in particular is the lemma in Appendix I. Although they only prove it for the orthogonal group, very similar arguments can be used to prove it for the symplectic group.