Trace-adapted orthonormal basis of two-by-two matrices ($I/\sqrt{2},\mathrm{diag}(1,-1)/\sqrt{2},(E_{12}+E_{21})/\sqrt{2},(E_{12}-E_{21})/\sqrt{2}$)

For the real two-by-two matrix space, the Frobenius inner product is $\mathrm{tr}(A^{t}B)={\sum }_{i,j}{a}_{ij}{b}_{ij}$. The displayed four matrices form an orthonormal basis. The first spans the identity direction, the next two span the traceless symmetric matrices, and the last spans the antisymmetric matrices. Symmetric and antisymmetric matrices are orthogonal because transposition changes the sign of their trace pairing; pairing with the identity gives the trace. This basis therefore displays the orthogonal direct sum into these three natural pieces.