Consider $P^n({\bf C})$ which is a quotient of $(S^{2n+1}, {\rm can})$.
If $ \{e_1, ... , e_n, Je_1, ... , Je_n\}$ is a basis on $T_xP^n({\bf C})$ where $J$ is an almost complex structure, then $$K(e_1,e_2)=1,\ K(e_1,Je_1)=4$$ where $K(x,y)={\rm Rm}(x,y,x,y)$
Note that curvature operator is symmetric :
$${\rm Rm } : \bigwedge^2 T_xM \times \bigwedge^2 T_xM \rightarrow {\bf R}$$
I want to find a basis on $ \bigwedge^2 T_xM$ which diagonalize ${\rm Rm}$.
How can we find ?
[Hint] I could find a fact that we must use the following (Problem 3.60 and its solution in the book Riemannian geometry - Gallot, Hulin, and Lafontaine) :
$$ \frac{\partial^2}{\partial s\partial t} [ K(x+sz,y+tw) - K(x+sw,y+tz) ] = 6\ {\rm Rm}(x,y,z,w)$$
And I think that the following would be used : $(-1)$-eignespace of $J$ is $$\{ e_i\wedge e_j - Je_i\wedge Je_j,\ e_i\wedge Je_j + Je_i \wedge e_j \}$$ and $1$-eignespace is $$\{ e_i\wedge e_j + Je_i\wedge Je_j,\ e_i\wedge Je_j - Je_i \wedge e_j,\ e_i\wedge Je_i \}$$