False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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Simple roots are pairwise orthogonal

Statement

False: distinct simple roots of a reduced crystallographic root system are generally not orthogonal; their inner product is nonpositive and can be nonzero.

Facts & Assumptions

Given: The standard model of $A_{2}$ and the notion of a simple root.

[L1]

For the root system $A_{2} = {e_{i} - e_{j} : 1 \leq i \neq j \leq 3}$ in the sum-zero subspace of $R^{3}$ , the roots $e_{1} - e_{2}$ and $e_{2} - e_{3}$ are simple with respect to the regular functional $x \mapsto (x, (3, 2, 1))$ (Existence of each classified root system, Positive systems and simple roots).

[L2]

Distinct simple roots satisfy $(α, β) \leq 0$ (Distinct simple roots have nonpositive inner product).

Refutation

technique · counterexample

1.1

In the model of [L1] take $α = e_{1} - e_{2}$ , $β = e_{2} - e_{3}$ ; the coordinates are $(1, - 1, 0)$ and $(0, 1, - 1)$ in the standard orthonormal basis of $R^{3}$ , so $(α, β) = 0 \cdot 1 + (- 1) \cdot 1 + 0 \cdot (- 1) = - 1 \neq 0$ .

L1algebra

2.1

Both $α$ and $β$ are simple roots by [L1], and they span a rank-two subsystem, so they are distinct simple roots that are not orthogonal; the negative value of their inner product is consistent with [L2]. This refutes the claim that simple roots are pairwise orthogonal.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)