classification – $mathfrak{su}(N)$ lie algebra

In the context of semisimple lie algebra, over $mathbb{k}$ algebraic closed and of characteristic zero, one can classifie all of then in $mathfrak{sl}_n(mathbb{k})$, $mathfrak{so}_{2n}(mathbb{k})$, $mathfrak{so}_{2n+1}(mathbb{k})$ and $mathfrak{sp}_{2n}(mathbb{k})$. My question is: which algebra corresponds to the $mathfrak{su}_n(mathbb{C})$?