I was recently taking some notes on the Cartan-Dieudonné theorem: if $(V,q)$ is a nondegenerate quadratic space of finite dimension $n$ over a field of characteristic not $2$, then every element of the orthogonal group $O(q)$ is a product of at most $n$ reflections through nondegenerate hyperplanes. (Moreover -- this is not the hard part! -- $n$ is sharp, as one sees by considering $-1$.)

So that's how things go in $O(q)$. At some point my mind wandered and I started thinking about $\operatorname{GL}(V)$ instead. Hyperplane reflections don't seem to have the same significance here, so I started to think about involutions -- i.e., elements of order $2$ -- instead. What is the subgroup of $\operatorname{GL}(V)$ generated by all the involutions?

Well, I was smart enough to realize that every involution has determinant $\pm 1$, so the subgroup in question is definitely contained in the subgroup of all matrices with determinant $\pm 1$ -- let's call this $\operatorname{GL}(V)^{(2)}$ -- hence usually proper. But is it actually this whole group? I checked one example: $\operatorname{GL}_2(\mathbb{F}_3)$ ~~is isomorphic to $S_4$~~ [not even close -- but still, one can see that my conclusion is true!], which is indeed generated by its involutions.

I didn't make much more progress than that, so I started googling. Eventually I came across the following paper:

Gustafson, W. H.; Halmos, P. R.; Radjavi, H. Products of involutions. Collection of articles dedicated to Olga Taussky Todd. Linear Algebra and Appl. 13 (1976), no. 1/2, 157–162.

Apparently in this paper the authors prove, among other things, that for any finite-dimensional vector space over any field, any element of $\operatorname{GL}(V)^{(2)}$ is a product of at most $4$ involutions. This is a pretty striking result, so I have some questions.

**Question 1**: How do you prove it? (I can't immediately access the paper, and unfortunately the description in the MathSciNet review was not immediately illuminating to me.)

**Question 2**: Can this result really not have been known until 1976?? The Cartan-Dieudonné theorem was proved at the latest in $1945$, the first publication date given for Dieudonné's book on classical groups. No one wondered about general linear groups for another $30$ years??

**Question 3**: In lots of generality we can take a group $G$ and ask what is the subgroup generated by its involutions. To fix ideas, suppose that $G$ is the group of $K$-rational points of a connected linear algebraic group over a field $K$. (Or choose whatever special case of this you like.) Can we say something about the subgroup generated by involutions? What about the least number $N$ of involutions so that every element of this subgroup is a product of at most $N$ involutions (if such an $N$ exists)?