The free Lie algebra functor is the left adjoint functor $FreeLieAlg$ to the forgetful functor $LieAlg\to Set$. The proof that a left adjoint exists relies on a concrete construction. Let $X$ be a set, then define recursively for $n\geq 1$ sets $X_n = \coprod_{p=1}^{n-1} X_p\times X_{n-p}$ with the basis of recursion $X_1 = X$. If $x\in X_p$ and $y\in X_q$ then denote $x.y$ the element $(x,y)\in X_{p+q}$; this defines a binary operation on $\coprod_{n=0}^\infty X_n$, which is therefore the free magma on the set $X$. Let $k$ be the ground ring (commutative and unital). As a $k$-module define $Lib_k(X) = k[\coprod_{n=0}^\infty X_n]$, the free $k$-module with basis $\coprod_{n=0}^\infty X_n$. It is a nonassociative $k$-algebra with product $(\sum_i a_i x_i).(\sum_j b_j y_j) = \sum_{i,j} a_i b_j (x_i.y_j)$ where both sums are finite and $a_i, b_j\in k$, $x_i,y_i\in X_i$. Define a two-sided ideal $I$ in this nonassociative $k$-algebra, generated by all elements of $Lib_k(X)$ of the form $a.a$ and all elements of the form $a.(b.c)+b.(c.a)+c.(a.b)$, where $a,b,c\in Lib_k(X)$. Then $FreeLieAlg(X) = Lib_k(X)/I$.
The free Lie algebra on the set $X$ is the result $FreeLieAlg(X)$ of evaluating the free Lie algebra functor on object $X$.
The subject of free Lie algebras is combinatorially rich with lots of open problems. By a 1953 theorem of A. I. Širšov (Shirshov) every Lie subalgebra of a free Lie subalgebra is free (an analogue of the Nielsen-Schreier theorem in combinatorial group theory). The study of bases of a free Lie algebra considered as a vector space is very nontrivial; special attention has been paid to so-called Hall bases.
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wikipedia: free Lie algebra
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sbseminar blog: Tannakian construction of the fundamental group and Kapranov’s fundamental Lie algebra
Nantel Bergeron, Muriel Livernet, A combinatorial basis for the free Lie algebra of the labelled rooted trees, Journal of Lie Theory 20 (2010) 3–15, pdf
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A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, pdf
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