Consider a (compact) Lie group $H$ that acts on its Lie algebra $\mathfrak h$ in the usual way, $x\mapsto gxg^{-1}$ for any $x\in\mathfrak h$ and $g\in H$. Suppose we are given a real symmetric bilinear form $\varphi$ on $\mathfrak h$, invariant under the action of the group, that is, $$ \varphi(gxg^{-1},gyg^{-1})=\varphi(x,y)\qquad \text{for all }x,y\in\mathfrak h\text{ and }g\in H. $$ Consider a Lie group $G$ such that $H$ is its subgroup. Under what conditions can the $H$-invariant form $\varphi$ on $\mathfrak h$ be extended to a $G$-invariant (real symmetric) form $\tilde\varphi$ on the Lie algebra $\mathfrak g$?

Here are some partial observations. $H$-invariance means that in a suitably chosen basis, the form $\varphi$ is represented by a matrix that commutes with all generators of $H$ in the adjoint representation. As long as $H$ is simple, Schur's lemma makes $\varphi$ essentially unique, $$ \varphi(x,y)=\mathrm{Tr}(xy) \tag{$\ast$} $$ up to a constant. It can then be trivially extended to the whole $\mathfrak g$. I can also construct examples where $\tilde\varphi$ does not exist. For instance, if $G$ is simple, then again by Schur's lemma, $\tilde\varphi$ would have to be given by ($\ast$) above, but when $H$ itself is not simple, I have more freedom in defining $\varphi$. In general, I expect $\varphi$ to be determined by a single real eigenvalue on every subspace of $\mathfrak h$, invariant under $H$. Once two such subspaces with different eigenvalues lie in the same invariant subspace of $G$, the extension $\tilde\varphi$ cannot exist. However, I would like to know if there is a more simple and precise criterion that would specify the (ideally necessary and sufficient) conditions under which the extension $\tilde\varphi$ exists.

Remark. This problem arises in different contexts in quantum field theory. For example, the most general $H$-invariant Yang-Mills Lagrangian takes the form $$ \mathcal L=-c_{ab}F^a_{\mu\nu}F^b_{\mu\nu} \tag{$\ast\ast$} $$ in the usual notation, where $c_{ab}$ is a symmetric tensor invariant under the adjoint representation of $H$ ($a,b$ are adjoint indices of $H$). What conditions must $c_{ab}$ satisfy so that this Lagrangian can be embedded into a Yang-Mills theory for the gauge group $G$?

A related question: under what conditions can ($\ast\ast$) be written in a matrix form, $$ \mathcal L=-\mathrm{Tr}(\Xi F_{\mu\nu}F_{\mu\nu}) $$ with $F_{\mu\nu}=F^a_{\mu\nu}T_a$ ($T_a$ being the generators of $H$), and a suitably chosen matrix $\Xi$? Note that when $H$ is simple, then necessarily $c_{ab}\propto\delta_{ab}$, and $\Xi$ can be taken proportional to the unit matrix.


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