Let $A,B$ be two subgroups of a group $G$. Let $\langle A,B\rangle$ be the group generated in $G$ by $A$ and $B$. Is it true that $A$ (or $B$) is a normal subgroup of $\langle A, B\rangle$?
My guess is that this is false, so let me ask what I am really interested in, although if the following is false I can further strengthen the hypothesis:
Let $A,B$ be two $Lie$ subgroups of a $Lie$ group $G$. Let $\langle A,B\rangle$ be the $Lie$ group generated in $G$ by $A$ and $B$. Is it true that $A$ (or $B$) is a normal subgroup of $\langle A, B\rangle$?
No. Take any simple group $G$ and any two non-trivial, proper subgroups $A,B$ that generate it. As a concrete example: take any Tarski monster group and any $\langle g\rangle, \langle h\rangle$ for two distinct and non-trivial elements $g,h$.
The same applies to Lie groups, since you can just enforce dimension $0$ on the previous example. If you want higher dimension then just take the product with $\mathbb{R}^n$.