Paired symmetric matrix decomposition?

Do any of the linear algebra gurus here know the name of or a citation for the following paired matrix decomposition: given symmetric matrices A and B, and an arbitrary vector v, find a decomposition of the form

A = W DA W'
B = W DB W'

for real matrices W satisfying W' v = [1,1,...,1] and DA and DB block-diagonal with 1x1 or 2x2 blocks. W is not required to be orthonormal.

I can probably figure out how to compute this myself (it seems very close to the generalized eigendecomposition), but if it already exists, Iā€™d rather acknowledge & call it by name.

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To answer my own question (I finally found the right combination of search terms), see Simultaneous tridiagonalization of two symmetric matrices.