TITLE: What is the basis for the Universal Enveloping Algebra of su(2)?
QUESTION [7 upvotes]: Given the standard basis for the Lie algebra $\mathfrak{su}(2)$ of SU(2), $\{i\sigma_1,i\sigma_2,i\sigma_3\}$ where
$\sigma_1=\Biggl(\begin{array}{cc}
0&1\\
1&0\end{array}\Biggr),\quad\sigma_2=\Biggl(\begin{array}{cc}
0&-i\\
i&0\end{array}\Biggr),\quad\sigma_3=\Biggl(\begin{array}{cc}
1&0\\
0&-1\end{array}\Biggr),$
I want to find a basis for the universal enveloping algebra, $\mathcal{U}(\mathfrak{su}(2))$. By the Poincare-Birkoff-Witt Theorem I believe we have
$\{i\sigma_1,i\sigma_2,i\sigma_3,-i\sigma_1\sigma_2,-i\sigma_1\sigma_3,-i\sigma_2\sigma_3,-i\sigma_1\sigma_2\sigma_3\}$,
in other words all lexicographically ordered monomials. However, since products of the Pauli matrices are Pauli matrices (ie $\sigma_1\sigma_2=i\sigma_3$) it would seem that the two algebras have the same basis, just with the Lie bracket $[,]$ replaced with matrix multiplication. Can someone tell me if this is correct?

REPLY [9 votes]: You are confused on a couple of points:
(1) The Poincare-Birkoff-Witt basis is the infinite set 
$$(i \sigma_1)^a (i \sigma_2)^b (i \sigma_3)^c \ \mbox{for} \ a,\ b,\ c,\ \geq 0.$$
You have only listed the cases where $a$, $b$ and $c$ are $0$ or $1$.
(2) The relation $\sigma_1 \sigma_2 = i \sigma_3$ does not hold in $U(\mathfrak{su}_2)$. That relation holds in the standard two dimensional representation of $\mathfrak{su}_2$, but it doesn't hold in (for example) the $3$ dimensional representation. The relations in $U(\mathfrak{su}_2)$ are those which hold in all representations of $\mathfrak{su}_2$. (Are you clear on what a representation of a Lie algebra means?)