We compute the v-decomposition numbers dλμ(v) for λ being a hook partition, and μ e-regular.

We first show that, under certain conditions, an ℓ-unital finite-dimensional Archimedean ℓ-algebra is isomorphic to a finite cyclic group ℓ-algebra with coefficients from a matrix ℓ-algebra. Then we characterize regular unital finite-dimensional Archimedean real ℓ-algebras which are either commutative or ℓ-reduced.