The classical Skitovich-Darmois theorem characterizes Gaussian distributions on the real line by the property that two distinct linear combinations of independent random variables are themselves independent. Extensions of this result to abelian groups are well understood, but the situation for non-abelian locally compact groups remains largely unresolved. The core problem is to determine precisely which non-abelian groups admit a meaningful analogue: that is, for which groups does the independence of two group-valued forms force each factor variable to follow a Gaussian-like or idempotent distribution, and what is the correct notion of Gaussian in this non-commutative setting.