In the following I consider complex manifolds and holomorphic vector bundles. I'm trying to understand the construction of non-trivial vector bundles $V$ via short exact sequences involving sums of line bundles, $$ 0 \to V \to \bigoplus_i L_{a_i} \to \bigoplus_j L_{b_j} \to 0 \,, $$ in arguably the simplest possible setting, of bundles over the complex projective line $\mathbb{CP}^1$.
It is known that any vector bundle on $\mathbb{CP}^1$ is isomorphic to a unique sum of line bundles. Hence given the two line bundle sums above, and assuming that the maps in the short exact sequence are as general as possible, one should be able to compute the line bundle sum to which $V$ is isomorphic. (Since a line bundle on $\mathbb{CP}^1$ is specified by a single integer, this is simply a map from two lists of integers of length $n_1$ and $n_2$ to one of length $n_1-n_2$.) Not all choices of input line bundle sums will result in a vector bundle $V$ - for some choices $V$ will not have constant rank but instead be a sheaf - and this too should be simple to determine from the integer lists specifying the input line bundle sums.
My question is whether there is any reference that discusses this problem or which would help me approach it. (There are certain obvious cases, but I would like to determine the complete result.)