Is the tangent sheaf $\mathscr{T}_{\mathbf{P}^2}$ a direct sum of line bundles? It is well known (theorem of Grothendieck) that every vector bundle on $\mathbf{P}^1$ is a direct sum of line bundles. What about $\mathbf{P}^2$? I figure the answer must be no, but is the tangent sheaf $\mathscr{T}_{\mathbf{P}^2}$ a counterexample? By the Euler sequence it shares most invariants with a direct sum of line bundles.

The total Chern class of $T_{\mathbb P^2}$ is $1 + 3h + 3h^2$ which is not a product of linear polynomials in $h$, so it cannot be the sum of two lines bundles. 
