A semiadditive category is preadditive exactly when every morphism has an additive inverse

Assume $\mathcal{C}$ is preadditive. [L2, L3] For parallel $f,g:A\to B$, product uniqueness gives $⟨f,g⟩=i_{1}f+i_{2}g$, and bilinearity gives $\mathrm{\nabla }⟨f,g⟩=f+g$. Thus the hom-group addition is compatible with the biproduct diagrams and equals the canonical law by [L3]. Every hom-set is an abelian group by [L2], so every morphism has an inverse for the canonical law.

Conversely, assume every morphism in $\mathcal{C}$ has an additive inverse for the commutative-monoid law from [L1]. Then every hom-set is a commutative monoid in which each element has an inverse, hence an abelian group. The bilinearity of composition is already part of [L1].

Therefore the hom-sets are abelian groups with bilinear composition exactly when every morphism has an additive inverse, which is the definition of preadditivity in [L2].