Define a Lie bracket on the group algebra of the permutation group $S_n$ in the following way: $$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$ where $\sigma, \tau \in S_n$, and the multiplication on permutations is defined as composition. My question is, what is the dimension of the Lie subalgebra generated by transpositions, i.e. $(ij)$? My conjecture is that the dimension is given by $C_n - \lfloor \frac{n}{2} \rfloor$, where $C_n$ is the Catalan number. Is this correct and what is the proof?
For example, when $n=3$, using the cycle notation, we have $$ [(12),(23)] = (132) - (123) \\ [(23),(31)] = (132) - (123) \\ [(31),(12)] = (132) - (123) \\ $$ and $$ [(12), (132) - (123)] = 2((23) - (13)), \text{etc.} $$ Therefore this algebra is $4 = C_3 - 1$ dimensional.