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I´d like to see the proof (I know it could be elemental) of this fact:

Let $L$ be a Lie algebra over a field $\mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x \in L$ if and only if$ [x,y]=−[y,x]$.

Thank you all

LH8
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    I'd like to see your work (I know it could be elemental) on this fact: Let $L$ be a Lie algebra over a field $\mathbb{F}$ with characteristic not 2. Then $[x,x]=0$ for any $x \in L$ if and only if$ [x,y]=−[y,x]$. – amWhy Jan 14 '19 at 19:49
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    ... and/or the source of the question (text (name and author and publication date)) and the subject you are studying, and your mathematical background, and/or an explanation as ti what motivated you to ask this question and how it is relevant to you and users on this site. Actually, please read How to ask a good question on math.se, as it well help you edit and improve this question after it is closed. – amWhy Jan 14 '19 at 19:53

2 Answers2

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I suppose that this not what you really want to ask, since the equivalence of those two conditions holds by the simple fact that both conditions hold for any Lie algebra.

However, for any algebra $(A,\star)$ over a field $F$ whose characteristic is not $2$, it is true that the conditions

  1. $(\forall x,y\in A):x\star x=0$;
  2. $(\forall x,y\in A):x\star y=-y\star x$

are equivalent. In fact, if the second conditions holds and if $x\in A$, then $x\star x=-x\star x$, which means that $2x\star x=0$. SInce the characteistic is not $2$, it follows from this that $x\star x=0$. And if the first condition holds, then, if $x,y\in A$,\begin{align}0&=(x+y)\star(x+y)\\&=x\star x+x\star y+y\star x+y\star y\\&=x\star y+y\star x\end{align}and therefore $x\star y=-y\star x$.

José Carlos Santos
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If $[x,x]=0$ for al $x$ then $0=[x+y,x+y]=[x,x]+[x,y]+[y,x]+[y,y]=[x,y]+[y,x],$ so $[x,y]=-[y,x].$

Conversely, if $[x,y]=-[y,x]$ for all $x,y$ then $[x,x]=-[x,x],$ so $2[x,x]=0.$ Since $F$ has characteristic not 2 then $[x,x]=0.$

positron0802
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