The PBW Theorem
In the literature there are many sources discussing the PBW-basis for Lie superalgebras, see for example M. Scheunert - The Theory of Lie Superalgebras theorem 1 and corollary in section 2.3
In a more recent source (Ian Musson - Lie superalgebras and enveloping algebras) the theorem is stated in the following form (for finite dimensional Lie superalgebras):
Theorem 6.1.2 (Musson) Let $x_1, \dots, x_m$ be a vector space basis for $\mathfrak{g}$ consisting of homogeneous elements. Then the set of all monomials of the form $$x_1^{a_1}\cdots x_m^{a_m}$$ with $a_i \in \mathbb{N}$ if $x_i \in \mathfrak{g}_0$ (i.e. $\bar{x_i}=0$) and $a_i\in \{0,1\}$ if $x_i \in \mathfrak{g}_1$ (i.e. an odd element) is a basis for $U(\mathfrak{g})$ over $\mathbb{C}$.
Question/Counterexample (?)
My question is on the following situation, which seems to disagree with the result above. Let $\mathfrak{g} = \mathfrak{sl}_{2|1}$ having even generators $E_1,F_1,H_1,H_2$ and odd generators $E_2,F_2$ subject to the following relations \begin{align*} [H_i,H_j] &= 0 = H_i H_j - H_j H_i\\ [H_i,E_j] &= a_{ij} E_j = H_i E_j - E_j H_i \\ [H_i, F_j] &= -a_{ij} F_j = H_i F_j - F_j H_i\\ [E_i, F_j] &= \delta_{ij} H_i = E_i F_j -(-1)^{\bar{ij}} F_j E_i \\ [E_1,[E_1,E_2]]&= 0 = E_1^2E_2 - 2 E_1 E_2 E_1 + E_2 E_1^2\\ [E_2,E_2] &= 0 = 2E_2^2 \\ [F_1, [F_1,F_2]] &= 0 = F_1^2F_2 - 2 F_1 F_2 F_1 - F_2 F_1^2 \\ [F_2,F_2] &= 0 = 2F_2^2 \end{align*} with $a_{ij}$ some integer coefficients (see e.g. Scheunert - Serre-type relations for special linear Lie superalgebras).
N.B. I have included the relations in $U(\mathfrak{sl}_{2|1})$ on the right side.
The theorem states that a basis is given by monomials $$H_1^a H_2^b E_1^c F_1^d E_2^e F_2^f$$ with $a,b,c,d \in \mathbb{N}$ and $e,f\in \{0,1\}$.
The question will maybe seem rather silly after such a build-up, but here goes:
Question: How does one express the element $E_2 E_1$ in terms of such a basis? (Similarly $E_2 E_1 E_2$)
As far as I can tell that is not possible and the basis is actually of the form $$H_1^a H_2^b E_1^c F_1^d (E_2E_1)^e E_2^f (F_2 F_1)^g F_2^h$$ with $a,b,c,d \in \mathbb{N}$ and $e,f,g,h\in \{0,1\}$
This basis seems to contradict with the theorem above. Thus if the answer on the first question is, `this is not possible'. What then should I make of the theorem above? Comments are most welcome.