Consider two real square matrices $A_1$ and $A_2$ and $t_1,t_2\in\mathbb{R}$. $A_1$ and $A_2$ do not commute. Consider the following matrix involving matrix trigonometric functions:

\begin{equation} M_1(t)=\begin{bmatrix} \cos(tA_1) & t\mathrm{sinc}(t A_1) \\ -A_1\sin(tA_1) & \cos(tA_1) \end{bmatrix} \end{equation} and $M_2(t)$ defined similarly by changing $A_1$ to $A_2$. Using the double-angle identities, it can be shown that

\begin{align} \Delta &= M_1(2t_1)-M_2(2t_2) \\ &= 2\begin{bmatrix} t_1\mathrm{sinc}(t_1A_1) & -t_2\mathrm{sinc}(t_2A_2) \\ \cos (t_1 A_1) & -\cos(t_2A_2) \end{bmatrix} \begin{bmatrix}-A_1\sin(t_1A_1) & \cos(t_1 A_1) \\ -A_2\sin(t_2A_2) & \cos(t_2 A_2) \end{bmatrix} \end{align}

which provides a factorization of the difference $\Delta$.

Is there a similar factorization for $M_1(2t_1)M_2(2t_2) - M_2(2t_4)M_1(2t_3)$ as the product of two matrices (or more)? What about the factorization of the more general case

$$\prod_{i=1}^k M_{\epsilon(i)}(2t_i) - \prod_{i=1}^k M_{\epsilon(i+1)} (2t_{2k-i+1})$$ with $\epsilon(i)=1$ if $i$ is odd and $2$ if $i$ is even?