  $L_{\infty}$ algebras describe the underlying algebraic structure of many
consistent classical field theories. In this work we analyze the algebraic
structure of Gauged Double Field Theory in the generalized flux formalism. The
symmetry transformations consist of a generalized deformed Lie derivative and
Double Lorentz transformations. We obtain all the non-trivial products in a
closed form considering a Generalized Kerr-Schild ansatz for the generalized
frame and we include a linear perturbation for the generalized dilaton. The
off-shell structure can be cast in an $L_{3}$ algebra and when one considers
dynamics the former is exactly promoved to an $L_{4}$ algebra. The present
computations show the fully algebraic structure of the fundamental charged
heterotic string and the $L^{\rm{gauge}}_{3}$ structure of (Bosonic) Enhanced
Double Field Theory.
