Split Grothendieck group of an additive category

Let $\mathcal{D}$ be an essentially small additive category (Additive category), and let $\mathrm{Iso}(\mathcal{D})$ be its set of isomorphism classes. Its split Grothendieck group is

$K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(\mathcal{D}):=\mathbb{Z}[\mathrm{Iso}(\mathcal{D})]/⟨e_{X\oplus Y}-e_X-e_Y⟩,$

where the free abelian group is generated by $\mathrm{Iso}(\mathcal{D})$ (Free abelian group on a set) and the subgroup is generated by the displayed elements for all pairs $X,Y\in \mathcal{D}$. Write $[X]$ for the image of $e_X$; then $[X\oplus Y]=[X]+[Y]$, including $[0]=0$. Only direct-sum relations are imposed; the isomorphism class of a biproduct is independent of its choice. In particular, for a finite-dimensional algebra $A$, write $K_0(A)$ for this group on the additive category of finite-dimensional projective left $A$-modules; it is distinct from the short-exact-sequence group of all finite-dimensional $A$-modules.