Lie bracket $[x,x]=0\implies [x,y]=-[y,x]$ Why does $[x,x]=0\implies [x,y]=-[y,x]$ in regard to Lie brackets.
I have tried to play around with bilinearity, but I can't get it to work.
$$[ax+by,z]=a[x,z]+b[y,z],[x,x]=0$$
I have tried subbing in $z=x$ and $z=y$, but I just can't obtain anti-commutativity. Thanks

$$[x+y,x+y]=0=[x,x]+[x,y]+[y,x]+[y,y]=[x,y]+[y,x]\Rightarrow [x,y]=-[y,x]$$
