Find endomorphisms $α$ and $β$ of $V$ satisfying the condition that $αβ$ is not nilpotent but $cα+dβ$ is nilpotent for all $c, d ∈ \mathbb{R}$ Let $V = \mathbb{R}^3$. Find endomorphisms $α$ and $F$ of $V$
satisfying the condition that $αβ$ is not nilpotent but $cα+dβ$ is nilpotent
for all $c, d ∈ \mathbb{R}$.
I tried $(cα+dβ)^2=c^2α^2+cd(αβ+βα)+d^2β^2=0$. But that don't seem to work, and I don't have another idea that guess and try.
Thanks.
 A: Hints


*

*Since $c \alpha + d \beta$ is nilpotent for all $c, d$, in particular $\alpha, \beta$ are nilpotent.

*Appealing to the Jordan Canonical Form gives that, up to similarity, there are only two nonzero nilpotent matrices in $M(3, \Bbb R)$, namely
$$\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix} \qquad \text{and} \qquad \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{pmatrix}.$$
A: I will denote by $A$, $B$ the desired endomorphisms. The condition $aA+bB$ nilpotent for all $a,b\in \mathbb R$ implies
$$
0=(aA+bB)^3 = (aA+bB)(aA+bB)(aA+bB) \\= a^3A^3 + a^2b( A^2B + ABA + BA^2) +
ab^2(B^2A+BAB+A^2B) + b^3 B^3,
$$
hence
$$
A^3 =0, \\
A^2B + ABA + BA^2=0, \\
B^2A+BAB+A^2B=0,\\
B^3=0.
$$
The first and the latter condition imply that $A,B$ have to be nilpotent.
Moreover, the condition $AB$ not nilpotent implies $A^2\ne0$ and $B^2\ne0$, otherwise the second or third line would yield $ABAB=0$.
Following Travis hint, you have to choose 
$$
A=\pmatrix{ 0 & 1 & 0\\ 0 & 0 & 1 \\ 0 & 0 & 0 },
$$
and $B$ similar to $A$. 

After fiddling around with the conditions on $B$ implied by $A^2B + ABA + BA^2=0$, I found one matrix $B$ that apparently does fulfill all the conditions
$$
B = \pmatrix{0&1&0\\
     1&0&1\\
     0&-1&0\\}.
$$
Still some high-level inside into this problem is highly appreciated :)
