On generation of the groups $GL_n(\mathbb{Z})$ and $PGL_n(\mathbb{Z})$ by three involutions, two of which commute

It is proved that the general linear group $GL_n(\mathbb{Z})$ (its projective image $PGL_n(\mathbb{Z})$ respectively) over the ring of integers $\mathbb{Z}$ is generated by three involutions, two of which commute, if and only if $n\geqslant 5$ (if $n=2$ and $n \geqslant 5$ respectively).