9.72 (1984)

Let $G_1$ and $G_2$ be Lie groups with the following property: each $G_i$ contains a nilpotent simply connected normal Lie subgroup $B_i$ such that $G_i/B_i \cong SL_2(\mathbb{K})$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. Assume that $G_1$ and $G_2$ are contained as closed subgroups in a topological group $G$, that $G_1 \cap G_2 \geqslant B_1B_2$, and that no non-identity Lie subgroup of $B_1 \cap B_2$ is normal in $G$. Can it then be shown (perhaps by using the method of “amalgams” from the theory of finite groups) that the nilpotency class and the dimension of $B_1B_2$ is bounded?