  Let ${\mathfrak g}$ be a complex simple Lie algebra with Borel subalgebra
${\mathfrak b}$. Consider the semidirect product $I{\mathfrak b}={\mathfrak
b}\ltimes{\mathfrak b}^*$, where the dual ${\mathfrak b}^*$ of ${\mathfrak b}$,
is equipped with the coadjoint action of ${\mathfrak b}$ and is considered as
an abelian ideal of $I{\mathfrak b}$. We describe the automorphism group
${\operatorname{Aut}}(I{\mathfrak b})$ of the Lie algebra $I{\mathfrak b}$. In
particular we prove that it contains the automorphism group of the extended
Dynkin diagram of ${\mathfrak b}$. In type $A_n$, the dihedral subgroup was
recently proved to be contained in ${\operatorname{Aut}}(I{\mathfrak b})$ by
Dror Bar-Natan and Roland Van Der Veen in arXiv:2002.00697 (where $I{\mathfrak
b}$ is denoted by $I{\mathfrak u}_n$). Their construction is handmade and they
ask for an explanation: this note fully answers the question.
