  We call an algebra $A$ commutator-simple if $[A,A]$ does not contain nonzero
ideals of $A$. After providing several examples, we show that in these algebras
derivations are determined by a condition that is applicable to the study of
local derivations. This enables us to prove that every continuous local
derivation $D\colon L^1(G)\to L^1(G)$, where $G$ is a unimodular locally
compact group, is a derivation. We also give some remarks on homomorphism-like
maps in commutator-simple algebras.
