Exceptional Siegel-Weil theorems for compact $\mathrm{Spin}_8$

Let $E$ be a cubic \'etale extension of the rational numbers which is totally real, i.e., $E \otimes \mathbf{R} \simeq \mathbf{R} \times \mathbf{R} \times \mathbf{R}$. There is an algebraic $\mathbf{Q}$-group $S_E$ defined in terms of $E$, which is semisimple simply-connected of type $D_4$ and for which $S_E(\mathbf{R})$ is compact. We let $G_E$ denote a certain semisimple simply-connected algebraic $\mathbf{Q}$-group of type $D_4$, defined in terms of $E$, which is split over $\mathbf{R}$. Then $G_E \times S_E$ maps to quaternionic $E_8$. This latter group has an automorphic minimal representation, which can be used to lift automorhpic forms on $S_E$ to automorphic forms on $G_E$. We prove a Siegel-Weil theorem for this dual pair: I.e., we compute the lift of the trivial representation of $S_E$ to $G_E$, identifying the automorphic form on $G_E$ with a certain degenerate Eisenstein series. Along the way, we prove a few more "smaller" Siegel-Weil theorems, for dual pairs $M \times S_E$ with $M \subseteq G_E$. The main result of this paper is used in the companion paper "Exceptional theta functions and arithmeticity of modular forms on $G_2$" to prove that the cuspidal quaternionic modular forms on $G_2$ have an algebraic structure, defined in terms of Fourier coefficients.