\cal O} _{A}{}^{B} = \delta_{A}{}^{B} - \eta P_{A}{}^{C} (R_{g})_{C}{}^{B} \ ,$$ in which we have explicitly introduced the deformation parameter $\eta$. Later we will restrict to the case $c=0$ in , in which case the parameter $\eta$ can be absorbed into the definition of $R$. The Yang-Baxter $\sigma$-model on a coset is given by [@Delduc:2014kha; @Matsumoto:2014nra] $${\cal L } = {\operatorname{Tr}}(J_+ P( {\cal O}^{-1}J_-) ) = J_{+}^{A} E_{AB} J_{-}^{B} \ , \quad E_{AB} = {\cal O}^{-1}{}_{B}{}^{C} P_{CA} \ .$$

Non-abelian T-duality Technology {#sec:natd}
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In this section we will mainly follow the approach of [@Sfetsos:2010uq; @Lozano:2011kb; @Itsios:2013wd] including the transformation of RR fluxes. Some subtleties are caused by the dualisation in a coset space and the approach here is slightly different to the one in [@Lozano:2011kb].

Let us consider the standard (bosonic) $\sigma$-model whose target is the coset space $F/G$ whose Lagrangian is given in eq. , and perform the non-abelian T-dual with respect to a subgroup $H \subset F$ (which need not, and in our applications mostly will not be, either semi-simple or a subgroup of $G$). Let $H_a$ be the generators of $\mathfrak{h}$ and $\tilde{H}^a$ generators of a dual algebra $\mathfrak{h}^\star$ normalised such that ${\operatorname{Tr}}(H_a \tilde{H}^b)= \delta^b_a$.

Let us we parametrise the coset representative as $g= h \hat{g}$. We define $\hat{J}= \hat{g}^{-1} d \hat{g}$ and $L = L^a H_a = h^{-1}dh$ such that $$J= \hat{J}+ L^a H_a^{\hat{g} }\ , \quad H_a^{\hat{g} } ={\operatorname{Ad}}_{\hat{g}^{-1}} H_a \ .$$ We also define $$\begin{aligned}
G_{ab} &= {\operatorname{Tr}}( H^{\hat{g}}_{a} P( H^{\hat{g}}_{b} ) ) \ , \quad
Q_{a} &= {\operatorname{Tr}}( \hat{J} P( H^{\hat{g}}_{a} ) ) \ .
\end{aligned}$$ In this notation the $H$ isometry of the target space is manifest since the metric corresponding to eq. is $$ds^2 = {\operatorname{Tr}}( \hat{J} P(\hat{J} )) + 2 Q^T L + L^T G L = {\operatorname{Tr}}( \hat{J} P(\hat{J} )) 