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Give the generator $p$ weight one and $g-1$ weight $\omega (g)$. For $\lambda \ge 0$, let ${\mathbb{Z}}_p[G{]}_{\lambda }$ be the ${\mathbb{Z}}_p$-span ofThe identitiesandtogether with the filtration inequalities show that the definition is independent of how an element is expanded and thatCompleteness and an Ordered basis of a complete p-valued group give separatedness through unique noncommutative power-series expansions. This is the Lazard filtration on a group algebra.

Multiplication by $p$ induces a central degree-one element $t$, so $\mathrm{gr}{\mathbb{Z}}_p[G]$ is a graded ${\mathbb{F}}_p[t]$-algebra. Sending the initial form of $g$ to the initial form of $g-1$ respects the Lie bracket, since the displayed algebra commutator has leading term represented by $[g,h]-1$. The universal property of the universal enveloping algebra therefore gives the Lazard enveloping-algebra mapIt is surjective because the defining filtered pieces are spanned by products of $p$ and the elements $g-1$.

For the principal congruence subgroup of ${\mathrm{SL}}_2({\mathbb{Z}}_p)$, let $e=E-1$, $f=F-1$, and $h=H-1$. In the group algebra,The factor $FE$ is congruent to one in filtration degree zero. Direct matrix multiplication, now applied to the group commutator, givesConsequently $ef-fe\equiv H^p-1\equiv p(H-1)=ph$ modulo ${\mathbb{Z}}_p[G{]}_3$. The same argument starts fromUsing $(1+p{)}^{-1}=1-p+p^2+O(p^3)$ in the two matrix commutators givesSince the prefactors $EH,FH$ may again be replaced by one at this precision, we obtain
Therefore

A bilinear form $B$ on a Lie algebra is invariant whenequivalently $B([x,y],z)+B(y,[x,z])=0$.

Choose a basis $(x_i)$ of $\mathfrak{g}$ and its $B$-dual basis $(x^i)$. The Casimir element iswhere $U(\mathfrak{g})$ is the universal enveloping algebra. The tensor ${\sum }_i{x}_i\otimes x^i$ corresponds under $B:\mathfrak{g}\cong {\mathfrak{g}}^{\ast }$ to the identity endomorphism, so invariance of $B$ makes it fixed by the diagonal adjoint action. Applying multiplication $U(\mathfrak{g})\otimes U(\mathfrak{g})\to U(\mathfrak{g})$ givesfor every $x\in \mathfrak{g}$. Thus $\Omega$ lies in the center of an associative algebra of $U(\mathfrak{g})$.

Let $D={\partial }_t$. Sincethe assignments ${\theta }_{\lambda }(y)=tD+\lambda$ and ${\theta }_{\lambda }(x)=D$ respect $[y,x]=-x$ and extend through the universal enveloping algebra.

The operator $y$ has distinct eigenvectors $t^n$ with eigenvalues $\lambda +n$, while $x(t^n)=n{t}^{n-1}$. Therefore the submodules are exactlyThe character $x\mapsto 0$, $y\mapsto \lambda$ has kernel $I_{\lambda }$, so $U(\mathfrak{g})/I_{\lambda }\cong k$ and $I_{\lambda }$ is maximal.

By part (a), the $S$-torsion in every module is a submodule. On $V=k\oplus kt$, $x(t)=1$ and $y$ has eigenvalues $\lambda ,\lambda +1$. If some $s\in S$ vanished on the $(\lambda +1)$-character, then $s(t)\in k$. Since $s(1)$ is the nonzero scalar given by its image modulo $I_{\lambda }$, subtracting a suitable constant from $t$ would produce an $S$-torsion vector $v$ with $xv=1$. Submodule closure would make $1$ torsion, contradicting $S\subseteq \mathcal{C}(I_{\lambda })$. ThusApply the same argument to each adjacent two-dimensional quotientwhere $x(t^{n+1})=(n+1)t^n\ne 0$ because $\mathrm{char}k=0$. Induction gives

An element $x$ of positive degree is primitive when $\Delta x=x\otimes 1+1\otimes x$. Graded commutators of primitives are primitive, so they form a graded Lie algebra. In a universal enveloping algebra over $\mathbb{Q}$, the primitive elements of a Hopf algebra are exactly its given Lie algebra: the associated graded algebra of the word-length filtration is symmetric, where no nonzero homogeneous polynomial of length greater than one is additive under the diagonal.

Choosing a positive system of a root system for a complex semisimple Lie algebra divides its root spaces into negative and positive ones. Their respective sums ${\mathfrak{n}}^{-}$ and ${\mathfrak{n}}^{+}$, together with the Cartan subalgebra $\mathfrak{h}$, give this direct sum of vector spaces. Both ${\mathfrak{n}}^{\pm }$ are nilpotent Lie algebras, and $\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ is a Borel subalgebra. The Poincaré-Birkhoff-Witt theorem turns this into an ordered vector-space decomposition of the universal enveloping algebra, useful for building Verma modules.