Does the tangent bundle of the projective space split?

Question

Does the tangent bundle of the projective space $\mathbb P^n$ over an algebraic closed field $k$ split i.e can be written as direct sum of two vector bundle of positive rank?

For cotangent bundle, the computation for the cohomology or hodge number shows it does not split as $h^{1,1}=1$. However, any higher cohomology of the tangent bundle vanishes.

Answer 1

Using Borel-Bott-Weil theorem it is easy to check that $$ Hom(T,T) \cong k. $$ This proves that $T$ is not split.