Is it true that the Weyl tensor vanishes on isotropic points?
In a riemannian manifold $(M,g)$, by Weyl tensor $W$ I mean $W=R-P \mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}} g$, where $R$ is the curvature tensor, $P$ the Schouten tensor given by $$ P=\frac{1}{n-2} \left( Ric- \frac{scal}{2(n-1)}g \right) $$ and $\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}$ the Kulkarni-Nomizu product. I read that if a point is isotropic (that is, the sectional curvature on that point is constant), then $W=0$. I wasn't able to prove it, any hint?