 Modified trace form on real matrices

On real $n$ by $n$ matrices with $n\ge 2$, split into traceless symmetric matrices, skew-symmetric matrices and scalar matrices. These summands are orthogonal for the displayed symmetric bilinear form. The form is positive on the first summand, negative on the second and negative on the scalar identity direction. Its inertia of a bilinear form is $(n(n+1)/2-1,n(n-1)/2+1,0)$, its rank is $n^2$ and its signature is $n-2$. Indeed $\mathrm{tr}(S^2)=\Vert S{\Vert }_F^2$, $\mathrm{tr}(K^2)=-\Vert K{\Vert }_F^2$ and $\varphi (tI,tI)=-n(n-1)t^2$.