Compatible Poisson brackets associated with 2-splittings and Poisson commutative subalgebras of $\mathcal S(\mathfrak g)$

Let ${\mathcal S}(\mathfrak g)$ be the symmetric algebra of a reductive Lie algebra $\mathfrak g$ equipped with the standard Poisson structure. If ${\mathcal C}\subset\mathcal S(\mathfrak g)$ is a Poisson-commutative subalgebra, then ${\rm trdeg\,}{\mathcal C}\le\boldsymbol{b}(\mathfrak g)$, where $\boldsymbol{b}(\mathfrak g)=(\dim\mathfrak g+{\rm rk}\mathfrak g)/2$. We present a method for constructing the Poisson-commutative subalgebra $\mathcal Z_{\langle\mathfrak h,\mathfrak r\rangle}$ of transcendence degree $\boldsymbol{b}(\mathfrak g)$ via a vector space decomposition $\mathfrak g=\mathfrak h\oplus\mathfrak r$ into a sum of two spherical subalgebras. There are some natural examples, where the algebra $\mathcal Z_{\langle\mathfrak h,\mathfrak r\rangle}$ appears to be polynomial. The most interesting case is related to the pair $(\mathfrak b,\mathfrak u_-)$, where $\mathfrak b$ is a Borel subalgebra of $\mathfrak g$. Here we prove that ${\mathcal Z}_{\langle\mathbb b,\mathbb u_-\rangle}$ is maximal Poisson-commutative and is complete on every regular coadjoint orbit in $\mathfrak g^*$. Other series of examples are related to decompositions associated with involutions of $\mathfrak g$.