Subalgebras, quotients, and finite products of nilpotent Lie algebras

Induction gives ${\gamma }_r(\mathfrak{h})\subseteq {\gamma }_r(\mathfrak{g})$ for all $r\ge 1$, because $[\mathfrak{h},A]\subseteq [\mathfrak{g},A]$. Thus any lower-series term vanishing in $\mathfrak{g}$ also vanishes in $\mathfrak{h}$.

If $\pi :\mathfrak{g}\to \mathfrak{g}\mathrm{/}\mathfrak{i}$ is the map in [L3], surjectivity and bracket preservation give ${\gamma }_r(\mathfrak{g}\mathrm{/}\mathfrak{i})=\pi ({\gamma }_r(\mathfrak{g}))$ by induction. Hence quotients inherit lower-series termination.

Componentwise bracketing in [L4] gives ${\gamma }_r({\prod }_{j\in J}{\mathfrak{g}}_j)={\prod }_{j\in J}{\gamma }_r({\mathfrak{g}}_j)$ for every $r\ge 1$. If $J$ is nonempty, let $c={\max }_{j\in J}\mathrm{cl}({\mathfrak{g}}_j)$. For $c=0$ every factor, and hence the product, is zero. For $c\ge 1$, the product's $(c+1)$st term is zero, while its $c$th term is nonzero in a factor attaining the maximum. Thus [L2] gives class exactly $c$ in either case. If $J$ is empty, [L4] identifies the product with zero and [L2] gives class zero.