  Let $M$ be a unital JB$^*$-algebra whose closed unit ball is denoted by
$\mathcal{B}_M$. Let $\partial_e(\mathcal{B}_M)$ denote the set of all extreme
points of $\mathcal{B}_M$. We prove that an element $u\in
\partial_e(\mathcal{B}_M)$ is a unitary if and only if the set
$$\mathcal{M}_{u} = \{e\in \partial_e(\mathcal{B}_M) : \|u\pm e\|\leq \sqrt{2}
\}$$ contains an isolated point. This is a new geometric characterisation of
unitaries in $M$ in terms of the set of extreme points of $\mathcal{B}_M$.
