Free Lie algebra

Remarks

The universal property determines $L(X)$ up to a unique isomorphism compatible with $\iota$. One construction: $L(X)$ is the Lie subalgebra generated by $X$ inside the free associative algebra $k\langle X \rangle$, with the commutator bracket.

Example: for $X = \set{x}$, the free Lie algebra is the line $k x$ with zero bracket. Check: a map $f$ sends $x$ to some $a \in \mathfrak g$, and the linear map $cx \mapsto ca$ is a homomorphism because $[cx, c'x] = 0$ and $[ca, c'a] = 0$. It is the only one, since a linear map on $k x$ is fixed by its value at $x$.

Non-example: for $X = \set{x, y}$, the abelian Lie algebra $k^2$ with $\iota(x) = (1, 0)$ and $\iota(y) = (0, 1)$ is not free on $X$. The map $f$ to $\mathfrak{sl}_2(k)$ with $f(x) = e$ and $f(y) = f_0$, where $e = E_{12}$ and $f_0 = E_{21}$, has no extension: a homomorphism would send $0 = [\iota(x), \iota(y)]$ to $[e, f_0] = E_{11} - E_{22} \ne 0$.

For $X = \set{x, y}$, the elements $[x, y]$, $[x, [x, y]]$ and $[y, [x, y]]$ are nonzero in $L(X)$ and linearly independent.