  We obtain a reduction of the vectorial Ribaucour transformation that
preserves the class of submanifolds of constant sectional curvature of space
forms, which we call the $L$-transformation. It allows to construct a family of
such submanifolds starting with a given one and a vector-valued solution of a
system of linear partial differential equations. We prove a decomposition
theorem for the $L$-transformation, which is a far-reaching generalization of
the classical permutability formula for the Ribaucour transformation of
surfaces of constant curvature in Euclidean three space. As a consequence, we
derive a Bianchi-cube theorem, which allows to produce, from $k$ initial scalar
$L$-transforms of a given submanifold of constant curvature, a whole
$k$-dimensional cube all of whose remaining $2^k-(k+1)$ vertices are
submanifolds with the same constant sectional curvature given by explicit
algebraic formulae. We also obtain further reductions, as well as corresponding
decomposition and Bianchi-cube theorems, for the classes of $n$-dimensional
flat Lagrangian submanifolds of $\mathbb{C}^n$ and $n$-dimensional Lagrangian
submanifolds with constant curvature $c$ of the complex projective space
$\mathbb C\mathbb P^n(4c)$ or the complex hyperbolic space $\mathbb C\mathbb
H^n(4c)$ of complex dimension $n$ and constant holomorphic curvature~4c.
