can take $n_a = 1$ for all the generators since $\deg([G_a, G_b])$ is then equal to one and fulfills (\[deg-g=n\]). The theorem then states that the rank is bounded by one, in agreement with (\[L-A-omega\]).

Assign the following degrees to $\eta^a$ and ${{\cal P}}_a$, $$\deg \eta^a = - n_a + 1, \; \deg {{\cal P}}_a = n_a - 1$$ If $\delta A = B$ and $\deg B = b$, then $\deg A = b - 1$ since $\delta$ increases the degree by one. Now ${{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} = G_a
\eta^a$ is of degree one. It follows that $[{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}] = [{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]_{\mbox{orig}}$ is of degree $\leq 1$ and hence, by (\[delta-om=d\]) and (\[d-p\]), $\deg {\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}
\leq 0$. More generally, one has $\deg {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k + 1$. Indeed, if this relation is true up to order $k-1$, then it is also true at order $k$ because in $$\delta {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \sim [{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (r)} \\ 