  Let $M_k^{(n)}$ be the space of Siegel modular forms of degree $n$ and even
weight $k$. In this paper firstly a certain subspace
$\mathsf{Spez}(M_k^{(2n)})$ the Spezialschar of $M_k^{(2n)}$ is introduced. In
the setting of the Siegel three-fold it is proven that this Spezialschar is the
Maass Spezialschar. Secondly an embedding of $M_k^{(2)}$ into a direct sum
$\oplus_{\nu = 0}^{\lfloor \frac{k}{10} \rfloor} \text{Sym}^2 M_{k + 2 \nu}$ is
given. This leads to a basic characterization of the Spezialschar property. The
results of this paper are directly related to the non-vanishing of certain
special values of L-functions related to the Gross-Prasad conjecture. This is
illustrated by a significant example in the paper.
