For Lie algebra $L$ with root $\alpha$, show that if $x \in L_{\alpha}$ then ad$x$ is nilpotent
121100
Rootspaces are $\mathop{ad}$ nilpotent
61868
Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.
12596
Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent iff $a_0,a_1,a_2,\ldots,a_n$ are nilpotent
70604
Lie algebra over $F$ of dimension $3$ whose derived algebra has dimension $1$ and lies in $Z(L)$
77722
Lie algebra of affine linear maps
128043
$A^n=0$ for $A$ nilpotent without the use of minimal polynomial
83595
Proving that $\mathbb{A}=\{\alpha \in \mathbb{C}: \alpha \text{ is algebraic over } \mathbb{Q} \}$ is not a finite extension
118396
Non abelian Lie group with abelian Lie algebra
22563
Units and Nilpotents
34183
An equivalent condition with $\{0\}$ being the only nilpotent ideal
70533
In what sense can the Lie algebra of $GL_n(\mathbb{C})$ be "identified" with $M_n(\mathbb C)$?
39657
Strictly Upper Triangular $n\times n$ Matrices form a Nilpotent Lie Algebra over $\mathbb{C}$ for $n\geq 2$
58653
The opposite of Weyl's theorem on Lie algebras
33423
Decomposition of semisimple Lie algebra via its roots
50236
Solvable Lie algebra with codimension 1 ideal
34908
$A$ is normal and nilpotent, show $A=0$
78263
Prove $GL_n(\Bbb{R})/SL_n(\Bbb{R}) \cong \Bbb{R}^\times$