Let $E\to\mathbb{CP}^1$ be a smooth real vector bundle, $\langle\cdot,\cdot\rangle$ a Riemannian metric on $E$, $\nabla$ a metric connection on $E$. In the complexfication $E\otimes\mathbb{C}$, $\langle\cdot,\cdot\rangle$ extends to a complex-bilinear form $(\cdot,\cdot)$ on $E\otimes\mathbb{C}$, and $\nabla=\nabla^{1,0}+\nabla^{0,1}$ splits into its $(1,0)$- and $(0,1)$-components. It is well-known that there is a unique holomorphic structure on $E\otimes\mathbb{C}$ so that $E\otimes\mathbb{C}\to\mathbb{CP}^1$ is a holomorphic vector bundle with $\nabla^{0,1}$ as its $\bar{\partial}$-operator. By a theorem of Grothendieck, $E\otimes\mathbb{C}$ splits holomorphically into a direct sum of holomorphic line bundles $L_1\oplus L_2\oplus\cdots\oplus L_m$. We order them according to their first Chern classes, i.e., $c_1(L_1)\geq\cdots\geq c_1(L_m)$ (where we in fact evalutate Chern classes on the fundamental class of $\mathbb{CP}^1$).
Question: Let $W_i$ be a meormorphic section of $L_i$. Then $c_1(L_i)+c_1(L_j)\neq0$ implies $(W_i,W_j)=0$.
Why is this true? This is a step from the 1988 annals paper Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes by M. J. Micallef & J. D. Moore. However, my knowledge of complex geometry is largely limited to the definitions, and I can't see why the above is true.
I need either a (detailed) explanation of why it is true or a book that helps me make sense out of it. Any help is appreciated!
Edit: In the context of the paper, $E$ is actually the pullback $f^*TM$ via some smooth map $f:\mathbb{CP}^1\to M$ where $M$ is a Riemannian manifold, and $\nabla$ is the pullback of the Levi-Civita connection on $M$.