I am studying Lie Algebras and after saw Engel's theorem my professor said that one can use Engel's theorem to compute the center of the following four Lie Algebras
$$ \mathfrak{gl}_{n}(\mathbb{F}),\mathfrak{sl}_{n}(\mathbb{F}), \mathfrak{so}_{n}(\mathbb{F}) \mbox{ and } \mathfrak{sp}_{n}(\mathbb{F}) $$
So I would like to ask if someone knows how to apply Engel's theorem in this case. I would be happy with any solution which find the center of the last two Lie Algebras.
Engel's theorem shows that the center of a nilpotent Lie algebra is always nontrivial. For the four Lie algebras we have other methods, which are better suited. Suppose that $F$ has characteristic zero. Then $\mathfrak{sl}_n(F)$ is a simple Lie algebra, so that its center, which is a proper ideal, has to be zero. For more details see this question. The same is true for the other simple Lie algebras $\mathfrak{so}_n(F)$ and $\mathfrak{sp}_n(F)$. The first Lie algebra, $\mathfrak{gl}_n(F)$ is reductive, and has $1$-dimensional center isomorphic to $F$, see here.
