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I would like to see the root system of type B_n emerge naturally from a maximal torus T in Spin(2n+1), without having to go through the Lie algebra. Since the spin group is simply connected the root lattice is just the kernel of an exponential homomorphism R^n -> T.

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Good game.

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Is it possible to have two linked circular cylinders in R^4? More specifically: are the two connected components of GL_2(R) linked in Mat_2(R)=R^4?

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Let A be a real matrix and let A*u=lambda*u where lambda=a+ib is a complex number and u=v+iw is a complex vector. Comparing real and imaginary parts of A(v+iw) = (a+ib)(v+iw) gives Av=av-bw and Aw=bv+aw, so that A stabilizes the real plane Rv+Rw.

If A is also orthogonal (A^T*A=I), it is easy to show that a=cos(theta) and b=sin(theta) for some angle theta. Furthermore, it is true (for generic theta) that the real vectors v,w are orthogonal, but I can't find the easy proof of this (there must be one).

Question: Why are v,w orthogonal?

If A is also orthogonal (A^T*A=I), it is easy to show that a=cos(theta) and b=sin(theta) for some angle theta. Furthermore, it is true (for generic theta) that the real vectors v,w are orthogonal, but I can't find the easy proof of this (there must be one).

Question: Why are v,w orthogonal?

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