The PBW theorem is stated in one of the following two forms (See Jacobson's or Humphreys text).
Let $L$ be a Lie algebra over $\mathbb{C}$ with a basis $\{x_i:i\in I\}$. Let $T(L)$ be the tensor algebra of $L$. For simplicity we write $x_i\otimes x_j$ as $x_ix_j$, keeping in mind that these are non-commuting variables.
Let $J$ be the ideal of $T(L)$ generated by $uv-vu-[u,v]$ for $u,v\in L$. Then $U=T(L)/J$ is universal enveloping algebra; let $u_i$ denote the image of $x_i\in L$ in $T(L)/J$.
Let $S$ be the polynomial ring $\mathbb{C}[z_i: i\in I]$ and $S_i$ be its subspace of polynomials of degree $\le i$
Then the PBW theorem is stated in one of the following ways:
(1) There is an action of $U$ on the space of polynomials $\mathbb{C}[z_i:i\in I]$ such that $$u_i.(z_{j_1}\ldots z_{j_r})=z_iz_{j_1}\ldots z_{j_r} \mbox{ if } i\le j_1\le \cdots \le j_r$$ $$u_i.(z_{j_1}\ldots z_{j_r})-z_i(z_{j_1}\ldots z_{j_r})\in S_{r-1}$$ $$ u_i.(u_j.(z_{j_1}\ldots z_{j_r}))=u_j.(u_i.(z_{j_1}\ldots z_{j_r})) + [u_i,u_j].(z_{j_1}\ldots z_{j_r})$$
(2) The monomials $\{ u_{i_1} u_{i_2}\cdots u_{is}: i_1\le i_2 \le\cdots \le i_s; s\ge 0\}$ form a basis of the vector space $U$.
Q. Is statement (1) better (stronger) than (2), considering their consequences?