Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles $$ \mathcal{P} = G\times_P \mathfrak{p} \subset G/P \times \mathfrak{gl} $$ and $$ \mathcal{T}_{G/P} = G\times_P \mathfrak{g/p}. $$

I would like to understand the space of sections of the latter, eventually in a way which works if $G$ is any principal bundle.

Bott showed that there is an exact sequence $$ 0\rightarrow H^0(G/P,\mathcal{P}) \rightarrow H^0(G/P,\mathcal{O}_{G/P}\otimes\mathfrak{g})\rightarrow H^0(G/P,\mathcal{P})\rightarrow H^1(G/P,\mathcal{P})\rightarrow 0 $$

Question 1: If $G/P = \mathbb{P}^{1}$ then $H^0(\mathbb{P}^{1}, \mathcal{T}_{\mathbb{P}^{1}}) = \operatorname{End}(\mathbb{C}^{2})/\mathbb{C} I$. How does $\mathcal{P}$ split as a direct sum of line bundles on $\mathbb{P}^1$?

Question 2: Is there a vanishing theorem for $H^1(G/P,\mathcal{P})$ for other flag varieties?

Related question: cohomology of tangent bundle