I'm trying do a exercise from Peter Petersen's book, but I don't know what do. Well, assume that
$$R=\frac{scal}{2n(n-1)}g\circ g+\left(Ric-\frac{scal}{n}g\right)\circ g+W$$
Where, R is the Riemannian curvature tensor, scal is the scalar curvature, W is the Weyl tensor, and we define the Kulkarni-Nomizu product of two (0,2)-tensor as a (0,4)-tensor:
$$h\circ k(v_1,v_2,v_3,v_4)=h(v_1,v_3)\cdot k(v_2,v_4)+h(v_2,v_4)\cdot k(v_1,v_3)-h(v_1,v_4)\cdot k(v_2,v_3)-h(v_2,v_3)\cdot k(v_1,v_4)$$
I know that this decomposition is orthogonal, in particular
$$|R|^2=\left|\frac{scal}{2n(n-1)}g\circ g\right|^2+\left|\left(Ric-\frac{scal}{n}g\right)\circ g\right|^2+|W|^2$$
I know that the curvature operator $\mathcal{R}:\Lambda^2TM\to\Lambda^2TM$ has the fllowing block decomposition:
$$\mathcal{R}=\begin{bmatrix} A&D \\ B&C \end{bmatrix}$$
where, $A,C$ are symmetrics and $D=B^*$
What I don't know is the following:
In dimension 4, show that $$|R|^2-\left|Ric-\frac{scal}{4}g\right|^2=tr(A^2-2BB^*+C^2)$$
Someone can help me? Thanks in advance!