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 ?



Nilpotent matrices with (Motzkin-Taussky) property L
The matrices $$A = \begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&0 \end{pmatrix}, B = \begin{pmatrix} 0&0&0 \\ 1&0&0 \\ 0&-1&0 \end{pmatrix}$$ satisfy $(sA+tB)^3=0$ for all $s,t$, but $$AB = \begin{pmatrix} 1&0&0 \\ 0&-1&0 \\ 0&0&0 \end{pmatrix},$$ which is not nilpotent, so $A$ and $B$ are not simultaneously triangularisable (as strictly upper triangular matrices).


