Timeline for A question about subspace in ${\bigwedge}^2({\mathbb R}^n)$

6 events

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Dec 7, 2020 at 18:25	comment	added	Noam D. Elkies		@alpoge: Fixed now, thanks again. Seems that there's no six-character requirement to edit your own answer!
Dec 7, 2020 at 18:23	history	edited	Noam D. Elkies	CC BY-SA 4.0	Correct the error noted by alpoge: h((1+X)^Y) is Y+h(X^Y), not X+h(X^Y), and so I need its orthogonality to Y, not X.
Dec 6, 2020 at 13:46	comment	added	Robert Bryant		Another way to see this (which, I think, is equivalent to your construction) is to consider the Lie algebra ${\mathfrak{spin}}(7)\subset{\mathfrak{so}}(8)\simeq\Lambda^2(\mathbb{R}^8)$. Every element of ${\mathfrak{spin}}(7)$ is conjugate to an element of its maximal torus, which is the same as the maximal torus of ${\mathfrak{su}}(4)={\mathfrak{spin}}(6)\subset {\mathfrak{spin}}(7)$. But clearly, the maximal torus of ${\mathfrak{su}}(4)\subset{\mathfrak{so}}(8)$ has no nonzero decomposable elements.
Dec 5, 2020 at 21:50	comment	added	Noam D. Elkies		Good catch, thanks -- yes, that's what I meant; will fix in the next edit (thanks too for the six-character warning).
Dec 5, 2020 at 21:38	comment	added	alpoge		(Apparently edits must be at least six characters! So instead I’ll comment: presumably it should be “$h\left((1+X)\wedge Y\right) = Y + h(X\wedge Y)$” and “... is orthogonal to $Y$”?)
Dec 5, 2020 at 18:35	history	answered	Noam D. Elkies	CC BY-SA 4.0