  We introduce a novel approach that employs techniques from noncommutative
Poisson geometry to comprehend the algebra of invariants of two $n\times n$
matrices. We entirely solve the open problem of computing the algebra of
invariants of two $4 \times 4$ matrices. As an application, we derive the
complete description of the invariant commuting variety of $4 \times 4$
matrices and the fourth Calogero-Moser space.
