$\ggo$, $\Aut({\mathfrak g })$. Let $\Dera({\mathfrak g })$ denote the subalgebra of $\Der({\mathfrak g })$ consisting of skew-symmetric derivations of $({\mathfrak g }, \la\,,\,\ra)$. Thus $\Dera({\mathfrak g })$ is the Lie algebra of the group of orthogonal automorphisms denoted by $\Auto({\mathfrak g })$: $$\Auto({\mathfrak g })=\{ \alpha \in \Aut({\mathfrak g }) : \la \alpha x, \alpha y \ra=\la x,y \ra \quad \forall x,y \in {\mathfrak g }\}.$$

For an arbitrary Lie algebra ${\mathfrak g }$, each $t \in \Aut({\mathfrak g })$ leaves invariant both the commutator ideal $C^1({\mathfrak g })$ and the center $\zz({\mathfrak g })$. Thus $t$ induces automorphisms of the quotient Lie algebras ${\mathfrak g }/\zz({\mathfrak g })$ and ${\mathfrak g }/C^1({\mathfrak g })$.

In particular if ${\mathfrak g }$ is solvable, ${\mathfrak g }/C^1({\mathfrak g })$ is a nontrivial abelian Lie algebra. Hence any $t\in\Aut({\mathfrak g })$ induces an element $s \in \Aut({\mathfrak g }/C^1({\mathfrak g }))$ which as an automorphism of an abelian Lie algebra, $s \in \GL(p,K)$ where $p = \dim( {\mathfrak g }/C^1({\mathfrak g }))$. Note that if ${\mathfrak g }$ is free nilpotent, $p$ coincides with the number of generators of ${\mathfrak g }$.

Below we proceed to the study in each case.

