# Rank of a Lie subgroup generated by two Lie subgroups

$$\DeclareMathOperator\rank{rank}$$Let $$G$$ be a compact connected Lie group and $$H$$, $$K$$ be two closed connected subgroups. By Mikhail Borovoi's answer to In a compact lie group, can two closed connected subgroups generate a non-closed subgroup?, $$L=\langle K, H\rangle$$ (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 $$\rank L=r.$$

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 $$H_2$$.)