  We prove that the predual, $M_*$, of a JBW$^*$-triple $M$ is a 1-Plichko
space (i.e. it admits a countably 1-norming Markushevich basis or,
equivalently, it has a commutative 1-projectional skeleton), and obtain a
natural description of the $\Sigma$-subspace of $M$. This generalizes and
improves similar results for von Neumann algebras and JBW$^*$-algebras.
Consequently, dual spaces of JB$^*$-triples also are 1-Plichko spaces. We also
show that $M_*$ is weakly Lindel\"{o}f determined if and only if $M$ is
$\sigma$-finite if and only if $M_*$ is weakly compactly generated. Moreover,
contrary to the proof for JBW$^*$-algebras, our proof dispenses with the use of
elementary submodels theory.
