manuscripta math. 83. 365 - 385 (1994) manuscripta
Generic 2 x 2 Matrices With Involution
Helmer Aslaksen and Eng-Chye Tan
Procesi has given a linear basis for the ring of m generic 2 X 2 matrices.
We do the same for the ring of m generic 2 X 2 matrices with transpose
Let k be a field of characteristic 0 and let M2 = M2( k) be the
set of 2 X 2 matrices over k. Consider an involution * on M2 of
the first kind. It follows from  that it is sufficient to study
the symplectic involution and the transpose involution. The case
of symplectic involution is essentially treated in Section 7 of ,
so we will only consider the transpose involution. We will use the
following notation: Mi is the set of symmetric matrices, M:; the
skew-symmetric matrices, Mf the matrices of trace 0, Mf+ the
symmetric traceless matrices and k the scalar matrices. Hence
M2 = Mi EB M2- = k EB Mf = k EB Mf+ EB M:;.