The title says it. Suppose I have a vector space $V$ equipped with a bilinear bracket such that $[x,y]=-[y,x]$, and define the universal enveloping algebra $U$ as usual: namely the tensor algebra on $V$ modulo the 2-sided ideal generated by
$x\otimes y-y\otimes x=[x,y]$
Then an ordered basis $\{x_i\}_{i\in I}$ of $V$ give rise to an order bases of $U$:
$\{\prod_{j=1}^tx_{i_j}^{k_j}\}_{i_1
Is this conclusion right?
I don't think the Jacobi identity is needed in the usual proof, but I just read one written by Paul Garrett (http://www.math.umn.edu/~garrett/m/algebra/pbw.pdf) which used it, so I'm not so sure about it.