Edit: I'm specializing this question to the compact case I'll ask about the noncompact case as a new question.

Let $ G $ be a compact connected semisimple Lie group. Do there always exist two finite order elements of $ G $ which generate a dense subgroup?

Example: $$ \frac{i}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} $$ and $$ \begin{bmatrix} \overline{\zeta_{16}} & 0 \\ 0 & \zeta_{16} \end{bmatrix} $$ generate a dense subgroup of $ SU_2 $.

This question is partially inspired by

Generating finite simple groups with $2$ elements

which shows that every finite simple group is 2-generated. Indeed even every finite quasisimple group is 2-generated Is every finite quasi-simple group generated by 2 elements? (however this fails for infinite simple groups: $ \mathrm{PSL}_2(\mathbb{Q}) $ is not 2-generated indeed not even finitely generated).

This is a cross-post of https://math.stackexchange.com/questions/4537024/dense-subgroups-generated-by-two-finite-order-elements an answer posted there points out that every compact semisimple Lie group can be topologically generated by 2 infinite order elements.

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