If $\{\alpha_1,\dots,\alpha_n\}$ is a basis of simple roots, so is $\{-\alpha_1,\dots,-\alpha_n\}$

Question

Let $\{\alpha_1,\dots,\alpha_n\}$ be a basis of simple roots for a root system $\Delta$. What is a simple explanation for why $\{-\alpha_1,\dots,-\alpha_n\}$ is another basis of simple roots?

Answer 1

One way of defining a collection $\{\alpha_1,\dots,\alpha_n\}\subseteq\Delta$ to be a set of simple roots is that every root $\alpha\in\Delta$ can be written uniquely as $\sum\limits_{i=1}^nk_i\alpha_i$, where either all $k_i$ are nonnegative integers or all $k_i$ are nonpositive integers.

From this, it's easy to see that $\{\alpha_1,\dots,\alpha_n\}$ is a set of simple roots if and only if $\{-\alpha_1,\dots,-\alpha_n\}$.

Another thing to note is that since $\Delta=-\Delta$ for any root system, the map $\alpha\mapsto-\alpha$ is an automorphism of the root system which clearly preserves the form. An automorphism of a root system always sends a collection of simple roots to another collection of simple roots.