On two letter identities in Lie rings

Let $L=L(a,b)$ be a free Lie ring on two letters $a,b.$ We investigate the kernel $I$ of the map $L\oplus L \to L $ given by $(A,B)\mapsto [A,a]+[B,b].$ Any homogeneous element of $L$ of degree $\geq 2$ can be presented as $[A,a]+[B,b].$ Then $I$ measures how far such a presentation from being unique. Elements of $I$ can be interpreted as identities $[A(a,b),a]=[B(a,b),b]$ in Lie rings. The kernel $I$ can be decomposed into a direct sum $I=\bigoplus_{n,m} I_{n,m},$ where elements of $I_{n,m}$ correspond to identities on commutators of weight $n+m,$ where the letter $a$ occurs $n$ times and the letter $b$ occurs $m$ times. We give a full description of $I_{2,m};$ describe the rank of $I_{3,m};$ and present a concrete non-trivial element in $I_{3,3n}$ for $n\geq 1.$