I'm presently reading through a paper of Shimura's; "Special Values of the Zeta Functions Associated with Hilbert Modular Forms". In the paper he defines $ \iota $ to be the main involution of $ M_{2}(F) $ and it's extension to $ M_{2}(F_{\mathbb{A}}) $, where $ F $ is a field and $ F_{\mathbb{A}} $ it's ring of ade`{e}les.
However I am not aware what this "main involution" is. The common use of the notation $ x^{-1} $ throughout the paper would suggest that he is not talking about the standard matrix inverse, and a quick google search suggests that this is in fact standard notation and assumed knowledge throughout the field. However I can't find the definition. I would much appreciate it if someone out there knows what the definition is.
It is $\pmatrix{a&b\cr c&d}\to \pmatrix{d&-b\cr -c&a}$, that is, $x\to {\rm tr}(x)-x$.
For a quaternion algebra, more generally, "trace" must be "reduced trace", which is consistent with the usual matrix trace.