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Every Lie algebra over a field $E$ gives rise to new Lie algebras over anysubfield $F \subseteq E$ by restricting the scalar multiplication. This paperstudies the structure of these underlying Lie algebra in relation to thestructure of the original Lie algebra, in particular the question how much ofthe original Lie algebra can be recovered from its underlying Lie algebra oversubfields $F$. By introducing the conjugate of a Lie algebra we show that insome specific cases the Lie algebra is completely determined by its underlyingLie algebra. Furthermore we construct examples showing that these assumptionsare necessary.  As an application, we give for every positive $n$ an example of a real$2$-step nilpotent Lie algebra which has exactly $n$ different bi-invariantcomplex structures. This answers an open question by Di Scala, Lauret andVezzoni motivated by their work on quasi-K\"ahler Chern-flat manifolds indifferential geometry. The methods we develop work for general Lie algebras andfor general Galois extensions $F \subseteq E$, in contrast to the originalquestion which only considered nilpotent Lie algebras of nilpotency class $2$and the field extension $\mathbb{R} \subseteq \mathbb{C}$. We demonstrate thisincreased generality by characterizing the complex Lie algebras of dimension$\leq 4$ which are defined over $\mathbb{R}$ and over $\mathbb{Q}$.

A note on the structure of underlying Lie algebras