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We show that to cubic order double field theory is encoded in Yang-Millstheory. To this end we use algebraic structures from string field theory asfollows: 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 gaugegroup and a `kinematic algebra' ${\cal K}$ that is a $C_{\infty}$-algebra. Thisstructure induces a cubic truncation of an $L_{\infty}$-algebra on the subspaceof level-matched states of the tensor product ${\cal K}\otimes \bar{\cal K}$ oftwo copies of the kinematic algebra. This $L_{\infty}$-algebra encodes doublefield theory. More precisely, this construction relies on a particular form ofthe Yang-Mills $L_{\infty}$-algebra following from string field theory or fromthe quantization of a suitable worldline theory.

The Gauge Structure of Double Field Theory follows from Yang-Mills Theory