Finite-dimensional Lie algebra

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$. A finite-dimensional Lie algebra over $\mathbb{F}$ is a finite-dimensional $\mathbb{F}$-vector space $\mathfrak{g}$ together with an $\mathbb{F}$-bilinear map

$[,]:\mathfrak{g}\times \mathfrak{g}⟶\mathfrak{g}$

such that, for all $X,Y,Z\in \mathfrak{g}$,

$[X,X]=0$

and

$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$

The first identity is alternation and the second is the Jacobi identity. Over $\mathbb{R}$ or $\mathbb{C}$, alternation is equivalent to skew-symmetry. Indeed, bilinearity and alternation give $0=[X+Y,X+Y]=[X,Y]+[Y,X]$, while skew-symmetry gives $2[X,X]=0$ and hence $[X,X]=0$ because both fields have characteristic zero. For a complex Lie algebra the bracket is required to be complex-bilinear, not merely real-bilinear.

The definition permits the zero bracket. The zero vector space, with its unique bracket, is a zero-dimensional Lie algebra. On a one-dimensional space, every alternating bilinear bracket is zero: any two vectors are scalar multiples of one vector and bilinearity reduces their bracket to $[X,X]=0$. A vector space is nonempty because it contains zero. No basis is selected by asserting finite-dimensionality, so the definition is choice-free. It is purely algebraic and has no metric, nondegeneracy, manifold-boundary, or endpoint condition.