I'm trying understand a computation on page 371 of Besse's book on <A HREF="http://www.amazon.com/Einstein-Manifolds-Classics-Mathematics-Arthur/dp/3540741208">Einstein Manifolds</A>.

I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form relative to the direct sum decomposition:
$$R=\begin{pmatrix}A&B\\C&D\end{pmatrix}$$
where $A=A^*,C=B^*,D=D^*$. And,
$$\begin{pmatrix}A&0\\0&D\end{pmatrix}-s/12=W,\mbox{the Weyl tensor}$$
The two components of the Weyl tensor $W^+=A-s/12,W^-=D-s/12$ are called the self-dual and the anti-self-dual parts respectively.

Now, pay attetion in this part:
>$$p_1(M)=-\frac{1}{8\pi^2}\int_MTr(R\wedge R)$$
where $R$ is considered as a matrix of 2-forms. Since $B$ and $B^*$ are acting on orthogonal spaces(this part is OK),
$$\begin{matrix}Tr(R\wedge R)&=&Tr(A\wedge A)+Tr(D\wedge D)\\&=&-2(|W^+|^2-|W^-|^2)\omega_g\end{matrix}$$
because $\alpha\wedge\alpha=|\alpha|^2\omega_g$ if $\alpha$ is sel-dual.

My questions are:

1) What $Tr(R\wedge R)$ stand for? 

2) How compute $Tr(R\wedge R)$, and why the term -2 appears in the expression?
 
I greatly appreciate any response, observation or correction!