  Weakly constrained double field theory, in the sense of Hull and Zwiebach,
captures the subsector of string theory on toroidal backgrounds that includes
gravity, $B$-field and dilaton together with all of their massive Kaluza-Klein
and winding modes, which are encoded in doubled coordinates subject to the
`weak constraint'. Due to the complications of the weak constraint, this theory
was only known to cubic order. Here we construct the quartic interactions for
the case that all dimensions are toroidal and doubled. Starting from the
kinematic $C_{\infty}$ algebra ${\cal K}$ of pure Yang-Mills theory and its
hidden Lie-type algebra, we construct the $L_{\infty}$ algebra of weakly
constrained double field theory on a subspace of the `double copied' tensor
product space ${\cal K}\otimes\bar{\cal K}$, by doing homotopy transfer to the
weakly constrained subspace and performing a non-local shift that is
well-defined on the torus. We test the resulting three-brackets, and establish
their uniqueness up to cohomologically trivial terms, by verifying the Jacobi
identities up to homotopy for the gauge sector.
