Are simple Lie algebras complete?

Question

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over a field of characteristic $p >0$. Is $\mathfrak{g}$ complete? If not, under what conditions is $\mathfrak{g}$ complete?

A Lie algebra is complete if its center is zero and all its derivations are inner.

Answer 1

Examples of simple modular Lie algebras in characteristic $p>2$ having outer derivations are, among others, the Block algebras and Frank algebras, see Seligman. Other examples are certain restricted simple modular Lie algebras of Cartan-type, given in the book by Strade and Farnsteiner.

There is a short argument by Zassenhaus, that a Lie algebra with non-degenerate Killing form has only inner derivations, over a field of arbitrary characteristic.

Answer 2

Not necessarily. However, simple Lie algebras with non-degenerate Killing forms are always complete.