$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$The free Lie algebra $\L(V)$ generated by an $r$-dimensional vector space $V$ is, in the
language of <https://en.wikipedia.org/wiki/Free_Lie_algebra>,
the free Lie algebra generated by any choice of basis $e\_1, \ldots , e\_r$ for the vector space $V$. (Work over the field ${\mathbb R}$ or ${\mathbb C}$, whichever you prefer.)
It is a graded Lie algebra  

$$\L(V) = V \oplus \L\_2 (V) \oplus \L\_3 (V) \oplus \ldots .$$
The general linear group $\GL(V)$ of $V$ acts on $\L(V)$ by gradation-preserving Lie algebra automorphisms.
Thus each graded piece $\L\_k (V)$ is a finite dimensional
representation space for $\GL(V)$. (The `weight' of $\L\_k (V)$ is $k$ in the sense that $\lambda \mathrm{Id} \in \GL(V)$ acts on $\L\_k (V)$ by scalar multiplication by $\lambda^k$.)
QUESTION: How does $\L\_k (V)$ break up into $\GL(V)$-irreducibles?


I only really know that $\L\_2 (V) = \Lambda ^2 (V)$, which is already irreducible.


To start the game off, perhaps some reader out there already is familiar with $\L\_3 (V)$
as a $\GL(V)$-rep, and can tell me its irreps in terms of the Young diagrams / Schur theory involving 3 symbols?


(My motivation arises from trying to understand some details of the subRiemannian geometry <https://en.wikipedia.org/wiki/Sub-Riemannian_manifold> of the Carnot group whose Lie algebra is the free $k$-step Lie algebra, which is $\L(V)$-truncated after step $k$. )



Breaking up the free Lie algebra into GL irreps
The Whitehouse module referred to in one of the other answers is not necessary, since it is related to the *cyclic* operad Lie, that is to the representation of $S\_{n+1}$ in $Lie(n)$.


The decomposition in terms of Young diagrams is, as far as I understand, first done in a paper of Kraskiewicz and Weyman (preprint W. Kraskiewicz, J. Weyman. Algebra of Invariants and the Action of a Coxeter Element. Math. Inst. Copernicus Univ. Chopina, Torun (1987), published W. Kraskiewicz, J. Weyman. Algebra of invariants and the action of a Coxeter element. Bayreuth. Math. Schr., 63 (2001)). Neither the preprint nor the published version are easily available, but lots of sources online discuss this topic. One set of slides that I found on Google right now which seems to contain all the statements you might possibly need is <https://personalpages.manchester.ac.uk/staff/Marianne.Johnson/Antalyaslides.pdf>


