We improve (cf. §5) on certain congruences for quadratic Stickelberger elements, as proved in [3]. For this we observe (cf. §4, th. 1) the existence of a distribution relation – additive with coefficients roots of unity – between simple quotients, introduced in §§2–3, of the Klein function of [12] for distinct complex lattices. This distribution relation, first noted by F.G. Frobenius [7], has an algebraic nature.