Let $G$ be a compact connected Lie group and $T$ is a maximal torus of $G$. Let $K$ be a non trivial connected Lie subgroup of $G$.


We say that $r \in \mathfrak{g}$ is a regular element of the Lie algebra $\mathfrak{g}$ if the stabilizer subgroup $G\_r$ of the adjoint action of $G$ on $\mathfrak{g}$ is a maximal torus of $G$.


We fix a maximal torus $T\_K$ of $K$.


2. Does there exist a regular element $r$ of both the Lie algebras $\mathfrak{g}$ and $\mathfrak{k}$ such that the stabilizer $G\_r$ is equal to $T$ and the stabilizer subgroup $K\_r$ is equal to $T\_K$ ?



Question about regular elements in a Lie subalgebra
It is possible to have a non-trivial, closed, connected Lie subgroup $K$ of a compact, connected Lie group $G$ such that no element of $\mathfrak k$ is regular in $\mathfrak g$. For example, consider $K = \operatorname{SU}\_2$ embedded in $G = \operatorname{SU}\_4$ as $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} a &&& b \\ & a & b \\ & c & d \\ c &&& d \end{pmatrix}$.


On second thought, even this is overkill; you could just take $K$ to be the (non-trivial) central torus in $G = \operatorname U\_2$. But maybe you wanted a non-central as well as non-trivial example.


