Let $\phi$ be an automorphism from a Lie algebra $L$ to itself. Is it necessary that $\phi(-x)=-\phi(x)$ for an element $x\in L$.
The automorphism must satisfy $[\phi(x),\phi(y)]=\phi([x,y])$. We may substitute $y$ by $-y$ to get $[\phi(x),\phi(-y)]=\phi(-[x,y])$. Still, I don't think we can get the relation $\phi(-x)=-\phi(x)$.
By definition, an automorphism of a Lie algebra L is a vector space automorphism $\varphi: L \rightarrow L$ such that $\varphi([x,y])=[\varphi(x),\varphi(y)]$. So one always has $\varphi(-x)=-\varphi(x)$.