= Solution
Use the degree-one convention implicit in this enveloping-algebra assertion: a <degree-one almost commutative algebra> has $F_0R=k$, a finite-dimensional $F_1R$ containing $1$ which generates $R$, and commutative <associated graded ring>. Taking the leading symbol of a <commutator> gives
$$
[F_iR,F_jR]\subseteq F_{i+j-1}R,
$$
in particular $[F_1R,F_1R]\subseteq F_1R$. Thus $\mathfrak g=F_1R$, equipped with $[u,v]=uv-vu$, is a finite-dimensional <Lie algebra>. Bilinearity and antisymmetry are immediate, and expanding the three nested <commutators> proves the <Jacobi identity>.
Its inclusion into the associative <algebra> $R$ is a <Lie algebra homomorphism>. The <universal enveloping algebra> is $T(\mathfrak g)$ modulo the relations $u\otimes v-v\otimes u-[u,v]$. Sending the tensor generators to their elements of $R$ kills these relations and gives an <algebra homomorphism> $U(\mathfrak g)\to R$. It is surjective because $F_1R$ generates $R$. Therefore
$$
\boxed{R\cong U(\mathfrak g)/\ker(U(\mathfrak g)\to R),\qquad \dim_k\mathfrak g<\infty.}
$$
The vector $1\in F_1R$ is retained as a central <Lie algebra> generator and maps to the associative identity; its corresponding relation in the kernel is that generator minus $1_{U(\mathfrak g)}$. One should not first replace $F_1R$ by $F_1R/k1$, since <commutators> of generators can be nonzero scalars.
Terminology is important here. Merely requiring a commutative finitely generated <associated graded ring> for an arbitrary weighted <filtration> is a broader definition of <almost commutative algebra>; it does not supply the finite-dimensional bracket-closed first step used in this argument. The degree-one assumptions make the requested enveloping presentation precise. The Noetherianity in part (b) also follows directly under the broader finite-generation convention.
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