Let $n,p,r \in {\mathbb Z}$ with $p \geq 2$, $r \geq 2$ and $n = p + r$.
Let $G \subseteq SU\left( n\right)$ be a Lie group and let $S \subset G$ be the subset of all elements in $G$ of the form $g = g_1 \oplus g_2$ where $g_1 \in SU\left( p\right)$, $g_2 \in SU\left( r\right)$.
Let $G_1 \subseteq SU\left( p\right)$ be the set of all $g_1$'s in $S$, i.e. ${G_1} = \bigcup\limits_{{g_1} \oplus {g_2} \in S} {\left\{ {{g_1}} \right\}} $.
Similarly, let $G_2 \subseteq SU\left( r\right)$ be the set of all $g_2$'s in $S$, i.e. ${G_2} = \bigcup\limits_{{g_1} \oplus {g_2} \in S} {\left\{ {{g_2}} \right\}} $.
If $G_1$ is a Lie group, does it follow that $G_2$ must also be a Lie group $\forall\; p,r$? If so, how would I go about proving this? If not, what would be a counterexample?
Does the answer change if $G_1 = SU\left( p\right)$?
I'm not sure how to proceed and any help would be much appreciated.