On Lie algebras with only inner derivations (Q1085259)

The author shows that for a finite dimensional Lie algebra L over any field F such that L/A is nilpotent where A is an abelian ideal, the following are equivalent: (1) All derivations of L are inner; (2) L is the direct sum of an abelian ideal A by a subalgebra B where \(B\simeq ad_ A(B)\) is a Cartan subalgebra of \(g\ell (A)\); (3) K being the algebraic closure of F, \(L^ K\) has a basis \(x_ 1,...,x_ n\), \(y_ 1,...,y_ n\) such that \([x_ i,y_ j]=\delta_{ij} x_ i\) and \([x_ i,x_ j]=[y_ i,y_ j]=0\) for \(i,j=1,...,n\).