$\DeclareMathOperator\rank{rank}$Let $G$ be a compact connected Lie group and $H$, $K$ be two closed connected subgroups. By Mikhail Borovoi's [answer](https://mathoverflow.net/a/62467) to [In a compact lie group, can two closed connected subgroups generate a non-closed subgroup?](https://mathoverflow.net/questions/62456/in-a-compact-lie-group-can-two-closed-connected-subgroups-generate-a-non-closed), $L=\langle K, H\rangle$ (the subgroup generated by $K$, $H$) is a closed connected subgroup of $G$ (hence a Lie subgroup). Now assume additionally that $$r=\rank(H\cap K)=\rank H=\rank K.$$ My question is if we can conclude that $$\rank L=r.$$



Rank of a Lie subgroup generated by two Lie subgroups
No. A simple example is to let $G = \mathrm{SO}(6)$, which has rank $3$. Let $v\_1$ and $v\_2$ be unit orthogonal vectors in $\mathbb{R}^6$ and let $H\_1\simeq \mathrm{SO}(5)\subset G$ be the stabilizer of $v\_1$ and let $H\_2\simeq \mathrm{SO}(5)\subset G$ be the stabilizer of $v\_2$. Then $H\_1\cap H\_2\simeq\mathrm{SO}(4)$ which has rank $2$, as do $H\_1$ and $H\_2$. However, $H\_1$ and $H\_2$ together generate $L = G$, which has rank $3$.


(Actually, come to think of it, one could do the same construction with $G = \mathrm{SO}(4)$ and get an even lower dimensional example in which the rank of $L$ is greater than that of $H\_1$ and $H\_2$.)


