I'm trying a exercise from Peter Petersen's book, and I did the following:
Let $*$ be the Hodge star operator, I know that $\Lambda^2TM$ decompose into $+1$ and $-1$ eigenspaces $\Lambda^+TM$ and $\Lambda^-TM$ for $*$.
I know that, if $e_1,e_2,e_3,e_4$ is an oriented orthonormal basis, then $$e_1\wedge e_2\pm e_3\wedge e_4\in\Lambda^{\pm}TM$$ $$e_1\wedge e_3\pm e_4\wedge e_2\in\Lambda^{\pm}TM$$ $$e_1\wedge e_4\pm e_2\wedge e_3\in\Lambda^{\pm}TM$$
What i can't prove is:
Thus, any linear map $L:\Lambda^2TM\to\Lambda^2TM$ has a block decomposition $$\begin{bmatrix} A&B \\ C&D \end{bmatrix}$$
$$A:\Lambda^+TM\to\Lambda^+TM$$ $$B:\Lambda^-TM\to\Lambda^+TM$$ $$C:\Lambda^+TM\to\Lambda^-TM$$ $$D:\Lambda^-TM\to\Lambda^-TM$$