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: $\operatorname{GL}_2(\mathbb{F}_3)$ <s>is isomorphic to $S_4$</s> [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:

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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. 

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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.

<b>Question 1</b>: 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.)

<b>Question 2</b>: 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??

<b>Question 3</b>: 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)?