  The goal of this article is to compute the Gerstenhaber bracket of the
Hochschild cohomology of the Fomin-Kirillov algebra on three generators over a
field of characteristic different from $2$ and $3$. This is in part based on a
general method we introduce to easily compute the Gerstenhaber bracket between
elements of $\operatorname{HH}^{0}(A)$ and elements of
$\operatorname{HH}^{n}(A)$ for $n \in \mathbb{N}_{0}$, the method by M.
Su\'arez-\'Alvarez to calculate the Gerstenhaber bracket between elements of
$\operatorname{HH}^{1}(A)$ and elements of $\operatorname{HH}^{n}(A)$ for any
$n \in \mathbb{N}_{0}$, as well as an elementary result that allows to compute
the remaining brackets from the previous ones. We also show that the
Gerstenhaber bracket of $\operatorname{HH}^{\bullet}(A)$ is not induced by any
Batalin-Vilkovisky generator.
