free Lie algebra

Fix a set $X$ and a commuative unital ring $K$. A free $K$-Lie algebra $𝔏$ on $X$ is any Lie algebra together with an injection $\iota :X\to 𝔏$such that for any $K$-Lie algebra $𝔤$ and function$f:X\to 𝔤$ impliesthe existance of a unique Lie algebra homomorphism $\widehat{f}:𝔏\to 𝔤$where $\iota \widehat{f}=f$. This universal mapping property is commonly expressedas a commutative diagram:

To construct a free Lie algebra is generally and indirect process.We begin with any free associative algebra $K⟨X⟩$ on $X$,which can be constructed as the tensor algebra over a free $K$-modulewith basis $X$. Then $K{⟨X⟩}^{-}$ is a $K$-Lie algebra with thestandard commutator bracket $[a,b]=ab-ba$ for $a,b\in K⟨X⟩$.

Now define $𝔉{𝔏}_K⟨X⟩$ as the Lie subalgebra of $K{⟨X⟩}^{-}$generated by $X$.

Theorem 1 (Witt).

[1, Thm V.7]$\mathrm{F}\mathit{}{\mathrm{L}}_K\mathit{}\mathrm{⟨}X\mathrm{⟩}$ is a free Lie algebra on $X$ and its universalenveloping algebra is $K\mathit{}\mathrm{⟨}X\mathrm{⟩}$.

It is generally not true that $𝔉{𝔏}_K⟨X⟩=K{⟨X⟩}^{-}$. For example, if $x\in X$ then $x^2\in K⟨X⟩$ but$x^2$ is not in $𝔉{𝔏}_K⟨X⟩$.