  We review some basic facts on vector fields, in the complex-analytic setting,
thus, obtaining a rationality result and an extension of the
Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex
manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$
the extended Lie algebra of infinitesimal automorphisms of $L$. If the
representation of $\mathfrak{g}_L$ on the space of holomorphic sections of $L$
is irreducible then $Z$ is rational; (2) Let $P$ be a holomorphic principal
bundle over the Riemann sphere, with structural group $G$ whose Lie algebra is
not equal to its nilpotent radical. Then there exists a Lie subgroup $H$ of $G$
which is a quotient of a Borel subgroup of ${\rm SL}(2)$ and such that $P$
admits a reduction to $H$.
