The basis for the canonical form of the trace of $2$ by $2$ matrix squared.
I have computed the trace of $A^2$ and its canonical representation:
$tr(A^2) = a_{11}^2+a_{22}^2+2a_{12}a_{21}=a_{11}^2+a_{22}^2+(a_{12}+a_{21})^2/2-(a_{12}-a_{21})^2/2$
How can one determine the basis of this canonical representation?
I think what you're looking for is to make each summand $1$ of the sum-of-squares formula while keeping the others $0$.
So that, in your case, we will get $$\pmatrix{1&0\\0&0},\quad \pmatrix{0&0\\0&1},\quad \pmatrix{0&\frac1{\sqrt2}\\ \frac1{\sqrt2}&0},\quad \pmatrix{0&-\frac1{\sqrt2}\\ \frac1{\sqrt2}&0}$$