In this paper, we show that there are infinitely many linearly independent elements in the abelianization of the Lie algebra of special derivations of a free Lie algebra by using the Morita traces. Furthermore, we show that the abelianization contains non-trivial elements which are killed by the Morita traces.

We would like to thank Takuya Sakasai, Yusuke Kuno, Masatoshi Sato for various comments for the the special derivation Lie algebras of free Lie algebras. In particular, we would like to thank Yusuke Kuno for communicating the non-finite generation of H₁(b₂,Z) with Soule elements. We also would like to thank the referee for his/her valuable comments for the original version of the paper. The second author is supported by JSPS KAKENHI Grant Number 25K06988.