$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$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 http://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 $k$. )

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$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 $k$. )

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 $\lambda Id \in Gl(V)$ 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 friends 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 http://en.wikipedia.org/wiki/Sub-Riemannian_manifold of the Carnot group whose Lie algebra is the free k-step Lie algebra, which is $L(V)$-trunated after step $k$. )

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$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 http://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 $k$. )