Setup: Let $E\rightarrow X$ be a holomorphic Hermitian vector bundle and $S\hookrightarrow E$ a holomorphic subbundle.
In Kobayashi's DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES, in proposition 1.6.14 it says that the second fundamental form vanish if the complementary $S^{\perp}$ is holomorphic and the isomorphism $$E=S\oplus S^{\perp}$$ holomorphic.
Now my question is, why is not it enough that $S^{\perp}$ is holomorphic? Is it possible that both $S$ and $S^{\perp}$ are holomorphic but the splitting is not? What can go wrong? In other words is there a situation where we cannot find a holomorphic retract to the map $S\rightarrow E$ ?