Lie algebras with fixed-point-free automorphisms are strongly unimodular; complex almost abelian ones admit them under a specific criterion, and complex filiform ones do so precisely when they are not characteristically nilpotent.
Jacobson:A note on automorphisms of Lie algebras
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Fixed-point-free automorphisms of solvable Lie algebras
Lie algebras with fixed-point-free automorphisms are strongly unimodular; complex almost abelian ones admit them under a specific criterion, and complex filiform ones do so precisely when they are not characteristically nilpotent.