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As far as I know, the index for H is a simple positive root (no negatives here), but the Index for E is a root (negatives allowed, no required of being simple). E.g. for $A_n$ there are n simple positive roots but the Lie algebra has dimension n^2-1! 2001:1716:4602:57C0:59CA:BED1:D7E0:5367 (talk) 10:52, 11 December 2023 (UTC)[reply]
