One of the consequences of the well-known Motzkin-Taussky theorem (https://www.jstor.org/stable/1990825) is the following : if two complex matrices A, B generate a vector space of diagonable matrices, then A and B commutes and in particular are simultaneously diagonable.

Does the result hold for nilpotent matrices : Let A and B two (complex) matrices such that sA+tB are nilpotent for all $s,t\in \mathbb{C}$ are they simultaneously triangularisable ?

One of the consequences of the well-known Motzkin-Taussky theorem (https://www.jstor.org/stable/1990825) is the following : if two complex matrices $A, B$ generate a vector space of diagonalisable matrices, then $A$ and $B$ commutes and in particular are simultaneously diagonalisable.

Does the result hold for nilpotent matrices : Let $A$ and $B$ two (complex) matrices such that $sA+tB$ are nilpotent for all $s,t\in \mathbb{C}$ are they simultaneously triangularisable ?