  Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{m\times n}$
denote the set of all $m\times n$ matrices over $\mathbb{H}$. Let
$\mathbf{i},\mathbf{j},\mathbf{k}$ be the imaginary quaternion units. For
$\eta\in\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, a square real quaternion matrix
$A$ is said to be $\eta$-Hermitian if $A^{\eta*}=A$ where $A^{\eta*}=-\eta
A^{\ast}\eta$, and $A^{\ast}$ stands for the conjugate transpose of $A$. In
this paper, we construct a simultaneous decomposition of four real quaternion
matrices with the same row number $(A,B,C,D),$ where $A=A^{\eta*}\in
\mathbb{H}^{m\times m}, B\in \mathbb{H}^{m\times p_{1}},C\in
\mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}}$. As applications of
this simultaneous matrix decomposition, we derive necessary and sufficient
conditions for some real quaternion matrix equations involving $\eta$-Hermicity
in terms of ranks of the coefficient matrices. We also present the general
solutions to these real quaternion matrix equations. Moreover, we provide some
numerical examples to illustrate our results.
