  Let $G_1 \times G_2$ be a subgroup of $\mathrm{SO}_3(\mathbb{R})$ such that
the two factors $G_1$ and $G_2$ are non-trivial groups. We show that if $G_1
\times G_2$ is not abelian, then one factor is the (abelian) group of order 2,
and the other factor is non-abelian and contains an element of order 2. There
exist finite and infinite such non-abelian subgroups.
