  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.
