2 I have modified the reference in the question.

Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, i=1\ldots n$ such that $u(x_i)=[x_i,a_i]$. The definition can be found in section 3 of Alekseev and Torossian's paper "The Kashiwara-Vergne conjecture and Drinfeld's associators"associators" arXiv:0802.4300v1.

My question is: why it has the name "tangential derivation", is there any historical reason or geometric intuition of it?

1

# About the term "tangential derivation" on a free Lie algebra.

Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, i=1\ldots n$ such that $u(x_i)=[x_i,a_i]$. The definition can be found in section 3 of "The Kashiwara-Vergne conjecture and Drinfeld's associators".

My question is: why it has the name "tangential derivation", is there any historical reason or geometric intuition of it?