The classic super Lie algebras are of type A: $\operatorname{sl}(m+1 \mid n+1)$, $m\neq n$, $\operatorname{psl}(n+1 \mid n+1)$, type B: $\operatorname{osp}(2m+1 \mid 2n)$, type C: $\operatorname{osp}(2 \mid 2n)$, type D: $\operatorname{osp}(2m \mid 2n)$.
But I find some paper they study $\operatorname{psu}(2, 2\mid 4)$. What type is this super Lie algebra? Thank you very much.
The classic Lie super-algebras listed are complex Lie algebras, i.e. no operation of conjugation is defined.
The $\mathfrak{psu}(2,2|4)$ is a real form of $\mathfrak{psl}(4,4)$, i.e. there exists an involution $\sigma: \mathfrak{psl}(4,4) \mapsto \mathfrak{psl}(4,4)$ (i.e. $\sigma^2 = \mathrm{id}$) and generators of $\mathfrak{psu}(2,2|4)$ are those linear combinations of generators, that are fixed by $\sigma$.