# Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras?

“In mathematics, a free Lie algebra, over a given field $K$, is a Lie algebra generated by a set $X$, without any imposed relations.” If we take the field $K$ to be the field $\mathbb{C}$ of complex numbers, what is the free algebra if

1. $X=\emptyset$?
2. $X=\{0\}$?
3. $X=\{0,1,2\}$?
4. $X=\mathbb{R}$?
5. $X=\mathbb{C}$?

#### Solutions Collecting From Web of "Examples of Free Lie Algebra"

So if, say, $X=\{1,2\}$, then the free Lie algebra $L(X)$, with $x\mapsto e_x$,
consists of all linear combinations of the elements $e_1,e_2$, $[e_1,e_2]$, $[e_1,[e_1,e_2]]$, $[e_2,[e_1,e_2]]$, $[e_1,[e_1,[e_1,e_2]]$, $[e_1,[e_2,[e_1,e_2]]$, $[e_2,[e_1,[e_1,e_2]]$, $[e_2,[e_2,[e_1,e_2]]$, and so forth, such that skew-symmetry and the Jacobi identity holds.

More formally, a free Lie algebra $L(x,y)$ on two letters $x,y$ has its elements in the ring of power series $K\langle x,y\rangle$ in the non-commuting variables $x,y$, where $K$ is a field.