This is a variant on the question posed [here](https://mathoverflow.net/questions/409899/is-every-matrix-involution-over-a-ufd-diagonalisable), in which the OP asks for a characterisation of the diagonalisable involutions in $\operatorname{GL}\_n(A)$, where $A$ is a $k$-algebra for some field $k$ of characteristic $\neq 2$. The accepted [answer](https://mathoverflow.net/questions/409899/is-every-matrix-involution-over-a-ufd-diagonalisable/410012#410012) shows that, if every finitely-generated projective module is free, then every involution is in fact diagonalisable.


Let us now restrict ourselves to the case where $A = k[x\_1,...,x\_l]$ is a polynomial ring over $k$. The Quillen-Suslin theorem says that all algebraic vector bundles over affine space are trivial, which translates to the condition that all finitely-generated projective modules over $A$ are free, so the solution above does apply. However, this feels to me like "killing a bee with a hand cannon" since the Quillen-Suslin theorem is quite a nontrivial result. Is there a more direct proof in this restricted setting?


As an aside, I'm hoping for a more explicit answer than just "use Quillen patching" as that will of course solve the problem, but I understand if that is the approach experts would take.



Order 2 matrices with entries in the polynomial ring over a field are diagonalisable
As abx pointed out in the comments, I misread the fact that this is an equivalence, so it is unlikely that there is a more elementary proof.


