I want show that when expressed in a PBW ordered basis, the Serre relations for a Lie Algebra become $0=0$. I am have trouble understanding what this question want. I think it just mean that Serre relations hold in universal enveloping algebra?
The Serre relations is $ad(E_{i})^{1-A_{ij}}E_{j}=ad(F_{i})^{1-A_{ij}}F_{j}=0$ for $i\neq j$, where $\{E_{i},F_{i},H_{j}:1\leq i\leq r\}$ are generators and $A$ is a Cartan matrix. So I think showing Serre relations become $0=0$ just mean showing $ad(E_{i})^{1-A_{ij}}E_{j}=0$ and same for $F_{i}$, is this right?
A PBW basis is a basis of Universal Enveloping Algebra $U(\mathfrak{g})$ for $\mathfrak{g}$ which consist of monomial of form $x_{1}^{k_{1}}...x_{n}^{k_{n}}$ where $x_{1}<...<x_{n}$ are elements of ordered basis of $\mathfrak{g}$.
I think I have to order basis consisting of $\{E_{i},F_{i},H_{j}:1\leq i\leq r\}$ and nested commutator, but not sure how.
I also know $A$ is cartan matrix so I find $1-A_{ij}\in\{1,2,3,4\}$.
Looking at $ad(E_{i})^{l}E_{j}$ for $l\in\{1,2,3,4\}$ and using $[x,y]=xy-yx$ in universal enveloping algebra I try to show this expression zero but do not manage.
Please clarify what the question means or give a hint, thankyou.