  We show that to cubic order double field theory is encoded in Yang-Mills
theory. To this end we use algebraic structures from string field theory as
follows: The $L_{\infty}$-algebra of Yang-Mills theory is the tensor product
${\cal K}\otimes \mathfrak{g}$ of the Lie algebra $\mathfrak{g}$ of the gauge
group and a `kinematic algebra' ${\cal K}$ that is a $C_{\infty}$-algebra. This
structure induces a cubic truncation of an $L_{\infty}$-algebra on the subspace
of level-matched states of the tensor product ${\cal K}\otimes \bar{\cal K}$ of
two copies of the kinematic algebra. This $L_{\infty}$-algebra encodes double
field theory. More precisely, this construction relies on a particular form of
the Yang-Mills $L_{\infty}$-algebra following from string field theory or from
the quantization of a suitable worldline theory.
