In B.C.Hall mathematics book on Lie Groups the Lie algebra is by definition closed under the Lie bracket operation, that is $[X, Y]\in \mathfrak g$ for every $X,Y \in \mathfrak g$.
The $\mathfrak su(2)$ algebra is defined as the $2 \times 2$ skew-Hermitian matrices with trace zero, and it's easy to see that $[X, Y]$ is skew-Hermitian, so $[X, Y] \in \mathfrak su(2)$.
However, in A.Zee Physics book on Group the $\mathfrak su(2)$ algebra is defined as the $2 \times 2$ Hermitian matrices with trace zero , and it's easy to see that $[X, Y]$ is skew-Hermitian, not Herimitian, so in this way $\mathfrak su(2)$ is not closed under the standard Lie bracket.
So to preserve the closure, it seems to me that the Lie bracket should be redefined as $[X,Y]\equiv -i(XY-YX)$. Since that is not mentioned at all in the A.Zee book I'm wondering if my arguments are correct.
