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. 



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> Where does the notion of split Grothendieck group originate, and why?
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This is mostly asked out of curiosity, but I'm also looking for further interesting examples.



Origin of notion of "split Grothendieck group"?
These groups are mentioned in [Swan '68 - Algebraic K-Theory, p.69]. He constructs $K\_0(\mathcal{A}, S)$ for a class $S$ of exact sequences in $\mathcal{A}$. Take the free abelian group mod the relations from sequences in $S$. For example the class of all exact sequences for the Grothendieck-group $K\_0(\mathcal{A})$ or all exact split seqences for the group you mentioned. The name 'split-Grothendieck-group' does not appear.


He generalizes further to $K\_0(\mathcal{A}, F)$ for a bifunctor $F:\mathcal{A} \times \mathcal{A} \to \mathcal{A}$ and obtains a generalized Picard-group.


