Lemma 1

For every integer $p \geq 0$, there exists a unique homomorphism $f_p$ of the K-module $g \otimes_K P_p$ into the K-module P satisfying the following conditions:

for $z_N \in P_{p-1}$. (The terms appearing in (Cp) are meaningful by (B_p).)*

Moreover, the restriction of $f_p$ to $g \otimes P_{p-1}$ coincides with $f_{p-1}$.

The last assertion follows from the others since the restriction of $f_p$ to $g \otimes_K P_{p-1}$ satisfies conditions (A_{p-1}), (B_{p-1}) and (C_{p-1}). We shall prove the existence and uniqueness of $f_p$ by induction on $p$. For $p = 0$, condition (A_0) gives $f_0(x_\lambda \otimes 1) = z_\lambda$ and conditions (B_0) and (C_0) are then obviously satisfied. Suppose now that the existence and uniqueness of $f_{p-1}$ are proved. We show that $f_{p-1}$ admits a unique extension $f_p$ to $g \otimes_K P_p$ satisfying conditions (A_p), (B_p) and (C_p).

We must define $f_p(x_\lambda \otimes z_M)$ for an increasing sequence M of $p$ elements.

If $\lambda \leq M$, the value is given by condition (A_p). Otherwise, M can be written uniquely in the form ($\mu, N$), where $\mu < \lambda, \mu \leq N$. Then

by (A_{p-1}), so that the left hand side of (C_p) is $f_p(x_\lambda \otimes z_M)$. Now the right hand side of (C_p) is already defined: for (B_{p-1}) allows us to write: where $w \in P_{p-1}$; hence the right hand side of (C_p) becomes:

Thus $f_p$ is defined uniquely and obviously satisfies conditions (A_p) and (B_p). Condition (C_p) is satisfied if $\mu < \lambda, \mu \leq N$. As $[x_\mu, x_\lambda] = -[x_\lambda, x_\mu]$, condition (C_p) is also satisfied for $\lambda < \mu, \lambda \leq N$. As (C_p) is trivially satisfied for $\lambda = \mu$, (C_p) is therefore satisfied if $\lambda \leq N$ or $\mu \leq N$. If none of these inequalities holds, $N = (\nu, Q)$, where $\nu \leq Q, \nu < \lambda, \nu < \mu$. Writing henceforth to abbreviate $f_p(x \otimes z) = xz$ for $x \in g$ and $z \in P_p$, we have by the induction hypothesis:

Now $z_\mu z_Q$ is of the form $z_\mu z_Q + w$, where $w \in P_{p-2}$. (C_p) can be applied to $x_\lambda(x_\nu(z_\mu z_Q))$, since $\nu \leq Q$ and $\nu < \mu$, and to $x_\lambda(x_\nu w)$ by the induction hypothesis, and hence to $x_\lambda(x_\nu(x_\mu z_Q))$. Hence:

Exchanging $\lambda$ and $\mu$ and subtracting:

and hence by the Jacobi identity

which completes the proof of Lemma 1.