= Solution
Choose a <basis> $x_1,x_2,x_3$ of the <Lie algebra> and filter its <universal enveloping algebra> by word length: $F_0R=\mathbb C$ and $F_dR$ is spanned by products of at most $d$ <basis> elements. The relations $x_jx_i-x_ix_j=[x_j,x_i]$ replace a degree-two <commutator> by degree one, so the degree-one symbols commute.
The <Poincaré-Birkhoff-Witt theorem> says the ordered <monomials> $x_1^{a_1}x_2^{a_2}x_3^{a_3}$ are a <basis>. Its relevant proof can be seen by reordering an inverted neighboring pair using the displayed relation. Reordering terminates because same-length terms have fewer inversions and bracket terms have smaller word length. Disjoint swaps commute; the only overlapping three-letter reorderings differ by $[x_i,[x_j,x_k]]+[x_j,[x_k,x_i]]+[x_k,[x_i,x_j]]=0$. The <Jacobi identity> therefore makes the reduced ordered expression independent of the reordering path. Ordered <monomials> have no remaining relations and form the <basis>. Their degree-$d$ symbols give
$$
\boxed{\operatorname{gr}R\cong\operatorname{Sym}(\mathfrak g)\cong\mathbb C[X_1,X_2,X_3]},\qquad\dim F_dR=\binom{d+3}{3}.
$$
For generators $m_1,\ldots,m_q$ of the <finitely generated module> $M$, put $M_d=\sum_jF_dRm_j$. This is a <good filtration>, and $\operatorname{gr}M$ is finite over the three-variable <polynomial ring>. Define the <dimension of a filtered module> as the <Krull dimension> of its <associated graded module>, equivalently the degree of the eventual cumulative <polynomial> $\dim_{\mathbb C}M_d$, or the <Gelfand–Kirillov dimension of a module>. Two <good filtrations> have bounded shifts, so the growth degree is independent of the generators. Since
$$
\dim M_d\le q\binom{d+3}{3},
$$
its <polynomial> degree is at most three. Therefore \b[every nonzero finitely generated $R$-module has dimension at most $3$]. The zero <module> may be assigned the usual separate empty-support convention. This bound also follows immediately from its <support of a module> inside affine three-space.
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