We prove two spectral identities. The first one relates the relative trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a weighted trace formula for $\GL_2$. The second relates a spherical variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are in accordance with a conjecture made by Jacquet, Lai, and Rallis, and are obtained without an appeal to a geometric comparison.

MSC Classifications:

11F70, 11F72, 11F30, 11F67 show english descriptions Representation-theoretic methods; automorphic representations over local and global fields
Spectral theory; Selberg trace formula
Fourier coefficients of automorphic forms
Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70 - Representation-theoretic methods; automorphic representations over local and global fields
11F72 - Spectral theory; Selberg trace formula
11F30 - Fourier coefficients of automorphic forms
11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

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