  We describe and characterize the contractively decomposable projections on
noncommutative $\mathrm{L}^p$-spaces. Our result relies on a new lifting result
for decomposable maps of independent interest and on some tools from ergodic
theory. Our theorem is new even for finite-dimensional Schatten spaces. Our
description allows us to connect this topic with $\mathrm{W}^*$-ternary rings
of operators and a slight generalization of our result for more general
projections makes $\mathrm{JBW}^*$-triples appear in this context. We also
prove that all rectangular $\mathrm{L}^p$-spaces associated with
$\mathrm{W}^*$-ternary rings of operators arise as contractively decomposable
complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. Finally, we
introduce a notion of $\mathrm{L}^p$-space associated to each $\sigma$-finite
$\mathrm{JBW}^*$-triple and we explain the link with the context of this paper.
