# Equivalent condition for base of a root system

In Humphrey's Lie algebra, following is the definition of base of a root system.

Let $\Phi$ be a root system in standard Euclidean space $\mathbb{R}^n$. A subset $\Delta$ of $\Phi$ is called as a base of $\Phi$ if

(1) $\Delta$ is basis of the vector space $\mathbb{R}^n$.

(2) Every $\alpha$ in $\Phi$ is a $\mathbb{Z}$-combination of elements of $\Delta$ with coefficients either all non-negative or all non-positive.

The elements of a base have property that their inner product is non-positive: $(\alpha,\beta)\leq 0$ for all $\alpha,\beta\in\Delta$ with $\alpha\neq\beta$.

Question: Suppose $\Delta'$ is a subset of $\Phi$ with condition that

(1) $\Delta'$ is a basis of the vector space $\mathbb{R}^n$.

(2) For any $\alpha\neq \beta$ in $\Delta'$, $(\alpha,\beta)\leq 0$. Then is $\Delta'$ necessarily a base of $\Phi$?

• Try to think about some examples first. What about $B_2$ and $G_2$? – Claudius Jan 25 '17 at 13:44
• OK, got it (ans: not true)! Actually I thought only of $A_2$ and $A_1\times A_1$; thanks for suggestion. – Beginner Jan 25 '17 at 14:57