We show that any sequence $\left({\phi }_n\right)$ of mutually orthogonal pure states on a JB algebra A such that $\left({\phi }_n\right)$ forms an almost discrete sequence in the relative topology induced by the primitive ideal space of A admits a sequence $\left(a_n\right)$ consisting of positive, norm one, elements of A with pairwise orthogonal supports which is supporting for $\left({\phi }_n\right)$ in the sense of ${\phi }_n\left(a_n\right)=1$ for all n. Moreover, if A is separable then $\left(a_n\right)$ can be taken such that $\left({\phi }_n\right)$ is uniquely determined by the biorthogonality condition ${\phi }_n\left(a_n\right)=1$. Consequences of this result improving...