Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda_{1},\cdots,\lambda_{r},\cdots,\lambda_{m}$ such that
$$B^{-1}A=\begin{pmatrix}
\lambda_{1}& 1&&&&&&\cr
&\lambda_{1}&\ddots&&&&&\cr
&&\ddots&&&\LARGE{0}&&\cr
&&&\lambda_{r}&1&&&\cr
&&&&\lambda_{r}&&&\cr
&&&&&\lambda_{r+1}&&\cr
&&\LARGE{0}&&&&\ddots&\cr
&&&&&&&\lambda_{m}
\end{pmatrix}$$
(*It is the canonical Jordan form*)
Can we ever find reals numbers $ t_ {1}, \cdots, t_ {p} $ so that the two following assertions are true:

- $A\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)B=B\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)A$
- $\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)B\quad\mbox{is nonsingular and diagonalizable }$?

**N.B :**

- The integer $p$ is not fixed.
- This question has arisen when studying the contollability of a real discrete-time nonlinear system. This explains why the matrices are supposed to be reals.

Thanks for help.