In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with *split* Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$ and writes $\langle \mathcal{A} \rangle$ for its split Grothendieck group: the free abelian group on objects $\langle A \rangle$ corresponding as usual to isomorphism classes, modulo sums$\langle C \rangle = \langle A \rangle + \langle B \rangle$ corresponding only to the situation $C \cong A \oplus B$.

This is a less familiar situation than the usual Grothendieck group with sums corresponding to short exact sequences which may or may not split.

Where does the notion of split Grothendieck group originate, and why?

This is mostly asked out of curiosity, but I'm also looking for further interesting examples.