I recently asked a question here about the paper "The curvature of 4-dimensional Einstein spaces." I got stuck again with the last theorem (2.2), where I get completely lost. They start the proof by saying:
Let $a=\lambda_{11}$ and $b=\lambda_{33}$ denote the minimum and maximum values of $\sigma$ and let $c=\lambda_{22}$ denote the real number such that: $a+b+c=\frac{1}{4}(\text{scalar curvature of} \ R)=\frac{1}{3}(\text{sum of all the critical values of} \ \sigma)$
Here comes the first doubt: How do they know that such a relation between the scalar curvature and the sum of all the critical values holds?
Then they say:
Then $a,b,c$ are critical values of $\sigma$ corresponding to a orthonormal critical plane basis $\{P_{11},P_{22},P_{33},P_{11}^{\perp},P_{22}^{\perp},P_{33}^{\perp} \}$ for $\Lambda^{2}.$
So, second doubt: How do they know that $c$ is critical from the assumptions made to pick it... and how do they know they have such an orthonormal basis?
After this, they go on:
Moreover, each of the nine planes $$P_{ij}=\frac{1}{2}\left( P_{ii}+P_{ii}^{\perp}+P_{jj}-P_{jj}^{\perp} \right)$$ are critical.
How do they know that? I would assume it is done by checking that $RP_{ij}=\lambda_{ij}P_{ij}+\mu_{ij}P_{ij}^{\perp},$ for certain real numbers $\lambda_{ij},\mu_{ij};$ but what they actually do is assume that $RP_{ij}$ has this form in order to compute $\mu_{ij},$ so there must be a more basic reason for the fact that these planes are critical.
Thanks a lot in advance for any help you may provide