Correspondence of Kubo-Ando Means over Real Division Algebras and Linearization of Means

In this paper, we establish a bijection between Kubo-Ando operator means defined on the cones $\mathscr{P}_{n}(\mathbb{H})$, $\mathscr{P}_{2n}(\mathbb{C})$, and $\mathscr{P}_{4n}(\mathbb{R})$. This correspondence is induced by the canonical embeddings relating quaternionic, complex, and real positive definite matrices. We investigate several structural and geometric properties preserved by these bijections, including compatibility with functional calculus, invariance under congruence transformations, and behavior with respect to natural metrics on these cones. As an application, we prove that every Kubo-Ando mean on $\mathscr{P}_{2}(\mathbb{D})$, where $\mathbb{D}\in\left\{\mathbb{R},\mathbb{C},\mathbb{H}\right\}$, admits an explicit affine expression in terms of the matrices involved. Using the embeddings above, we derive explicit formulas for operator means on special classes of real $4\times4$ positive definite matrices arising as images of the cones $\mathscr{P}_{2}(\mathbb{C})$ and $\mathscr{P}_{2}(\mathbb{H})$. In particular, we obtain trace-determinant formulas for the geometric mean in the real, complex, and quaternionic settings.