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.