Conceivably it is useful to imbed Siegel's result in a somewhat larger context, so that it is a bit of a special case of something. Namely, (and this is still a special case...), even-sized orthogonal groups $O(Q)$ defined over a number field $k$ "pair" with $Sp(2n,k)$ for all choices of size $2n$, as mutual commutators inside a two-fold cover $Mp(2n\cdot \dim Q)$ of $Sp(2n\cdot \dim Q)$. The "Segal-Shale-Weil/oscillator" repn of the adele group (restricted from a repn of the metaplectic Mp) gives a well-defined mapping from repns (local or global) from irreducibles of $O(Q)$ to irreducibles of $Sp(2n)$ for $2n\gg \dim Q$, and in the other direction for the opposite inequality. (The precise cut-offs are about "first-occurrence", as in work of Kudla-Rallis and others.)

In a rather degenerate situation, for $\dim Q\gg 2n$, the constant $1$ on the adelic orthogonal group is mapped to some sort of automorphic form on $Sp(2n)$. It is not profoundly difficult, but non-trivial, to see that the image is the Siegel-type Eisenstein series. With sufficiently many decades of hindsight, and with the benefit of working on adele groups rather than classically, as Siegel did, various people have found simplified arguments: there is a bibliography and an example of a simplified argument on-line in http://www.math.umn.edu/~garrett/m/v/siegel_weil.pdf, also in the Eisenstein Series volume from AIM Harris-Skinner-Lie have a wider-scope discussion in his article "A simple proof of rationality of Siegel-Weil" (http://www.math.jussier.fr/~harris/resarticles/SW.pdf)

The translation into classical language engenders the discussion of lattices in a genus and cardinalities of automorphism groups, much as comparison of idele class groups to ideal class groups and unit groups gives rise to classical details regarding the latter.

Conceivably it is useful to imbed Siegel's result in a somewhat larger context, so that it is a bit of a special case of something. Namely, (and this is still a special case...), even-sized orthogonal groups $O(Q)$ defined over a number field $k$ "pair" with $Sp(2n,k)$ for all choices of size $2n$, as mutual commutators inside a two-fold cover $Mp(2n\cdot \dim Q)$ of $Sp(2n\cdot \dim Q)$. The "Segal-Shale-Weil/oscillator" repn of the adele group (restricted from a repn of the metaplectic Mp) gives a well-defined mapping from repns (local or global) from irreducibles of $O(Q)$ to irreducibles of $Sp(2n)$ for $2n\gg \dim Q$, and in the other direction for the opposite inequality. (The precise cut-offs are about "first-occurrence", as in work of Kudla-Rallis and others.)

In a rather degenerate situation, for $\dim Q\gg 2n$, the constant $1$ on the adelic orthogonal group is mapped to some sort of automorphic form on $Sp(2n)$. It is not profoundly difficult, but non-trivial, to see that the image is the Siegel-type Eisenstein series. With sufficiently many decades of hindsight, and with the benefit of working on adele groups rather than classically, as Siegel did, various people have found simplified arguments: there is a bibliography and an example of a simplified argument on-line in http://www.math.umn.edu/~garrett/m/v/siegel_weil.pdf, also in the Eisenstein Series volume from AIM Harris-Skinner-Li have a wider-scope discussion in "A simple proof of rationality of Siegel-Weil" (http://www.math.jussier.fr/~harris/resarticles/SW.pdf)

The translation into classical language engenders the discussion of lattices in a genus and cardinalities of automorphism groups, much as comparison of idele class groups to ideal class groups and unit groups gives rise to classical details regarding the latter.