1$, the equations of motion of the tensor fields do not follow from an action. We can still consider the `pseudo-action'~\cite{Ferrara-ml-1997gh,Riccioni-ml-2001bg} for the bosonic fields\footnote{To avoid confusion with standard notation $H$ for harmonic forms, we do not follow the notation of~\cite{Ferrara-ml-1997gh,Riccioni-ml-2001bg} for the three-forms and the kinetic matrix. To convert, use $G^r = H^r_{theirs}$ and ${\cal M}_{rs} = (G_{rs})_{theirs}$.} \be \mathcal{L} = \frac14 R - \frac12 \partial_\mu v_r \partial^\mu v^s - \frac{1}{12} {\cal M}_{rs}G^r_{\mu\nu\rho}G^{s\,\mu\nu\rho},\label{eq:6dsugra_more_tensors} \ee that captures the equations of motion of the scalar fields and the metric. The scalars parametrize the coset $\SO(1,n_T)/\SO(n_T)$. They can be organized in the $\SO(1,n_T)$-matrix $V = \Big(\begin{smallmatrix}v_r\\x_r^M\end{smallmatrix}\Big)$ with $M = 1\ldots n_T$ and $r=0\ldots n_T$.\footnote{It is customary to write the $\SO(1,n_T)$ conditions $V\eta V^T= V^T \eta V = \eta$ in component notation as $v_r v^r = 1, v^r x^M_r=0, v_r v_s - x_r^M x_s^M = \eta_{rs}$.} They enter the tensor dynamics through the scalar metric ${\cal M} = \eta V^T V \eta$, with $\eta$ the $\SO(1,n_T)$-metric, or in index notation \be {\cal M}_{rs} = v_r v_s + x_r^M x_s^M\,. \ee The dynamics of the $n_T +1$ tensor fields $G^r$ are captured by the self-duality conditions and Bianchi identities \be {\cal M}_{rs} G^s = \eta_{rs} \star G^s\,,\qquad dG^r = 0\,,\label{eq:3form_duality} \ee where $\star$ is the six-dimensional Hodge star operator. Finally, the Einstein equations are: \be R_{\mu\nu}=2\partial_\mu v^r\partial_\nu v_r+\frac 12 {\cal M}_{rs}G_{\mu\alpha\beta}^rG_{\nu}^{s\hspace{2pt}\alpha\beta}\,.\label{eq:EE_moretensors} \ee ]]>