So I feel like asking the following likely openended question: What good generalizations of the notion of Cartan subspace do we have?
To be precise, let $G\curvearrowright V$ be an algebraic representation over a field $k$ satisfying some conditions you like (e.g. $\mathrm{char}(k)=0$, $G$ is connected reductive). When can we call a subspace $S\subset V$ a Cartan subspace with a good reason? (hopefully generalizing the notion of Vinberg as duplicated below, at least that of Cartan subalgebras for adjoint representations for sure)
My motivation is that I am trying to generalize some work of myself about affine Springer fibers and possibly a bit Springer theory. I end up deciding that the notion of Cartan subspace is essential.

When $\mathrm{char}(k)=0$, $H$ is a reductive group, $\theta$ is an automorphism on $H$ of finite order $m$, $\zeta_m\in k$ is a primitive $m$th root of unity, $G:=(H^{\theta})^o$, $V:=\mathfrak{h}^{\theta=\zeta_m}$ and $G\curvearrowright V$ comes from the original adjoint representation $H\curvearrowright\mathfrak{h}$, Vinberg defined a Cartan subspace to be a maximal subspace consisting of commuting semisimple elements in $V$. Here commuting means commuting in $\mathfrak{h}$, and semisimple means its $G$orbit is closed, or equivalently it is semisimple in $\mathfrak{h}$. Vinberg proved that all Cartan subspaces are conjugate, generalizing previous results for symmetric spaces when $m=2$.