Lemma 2

Let B and C be two lower triangular matrices of orders m and n respectively, with diagonals $(\beta_i)_{1 \leq i \leq n}$ and $(\gamma_j)_{1 \leq j \leq n}$. Let us identify the lexicographic product of the ordered sets $\{1, 2, \ldots, m\}$ and $\{1, 2, \ldots, n\}$ with the interval $\{1, 2, \ldots, mn\}$. Then the tensor product matrix (II, p. 357) B $\otimes$ C is lower triangular with diagonal $(\beta_i \gamma_j)$.

This follows immediately from the definition of the tensor product of two matrices (loc. cit.) and of the lexicographic product (Set Theory, III, p. 157).