In the last years algebraic tools have been proven to be useful in
phylogenetic reconstruction and model selection by means of the study of
phylogenetic invariants. However, up to now, the models studied from an
algebraic viewpoint are either too general or too restrictive (as group-based
models with a uniform stationary distribution) to be used in practice.
In this paper we provide a new framework to work with time-reversible models,
which are the most widely used by biologists. In our approach we consider
algebraic time-reversible models on phylogenetic trees (as defined by Allman
and Rhodes) and introduce a new inner product to make all transition matrices
of the process diagonalizable through the same orthogonal eigenbasis. This
framework generalizes the Fourier transform widely used to work with
group-based models and recovers some of the well known results. As
illustration, we exploit the combination of our technique with algebraic
geometry tools to provide relevant phylogenetic invariants for trees evolving
under the Tamura-Nei model of nucleotide substitution