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In my discussion of the Oppenheim conjecture in my recent post on Ratner’s theorems, I mentioned in passing the simple but crucial fact that the (orthochronous) special orthogonal group $SO(Q)^+$ of an indefinite quadratic form on ${\Bbb R}^3$ can be generated by unipotent elements. This is not a difficult fact to prove, as one can simply diagonalise Q and then explicitly write down some unipotent elements (the magic words here are “null rotations“). But this is a purely algebraic approach; I thought it would also be instructive to show the geometric (or dynamic) reason for why unipotent elements appear in the orthogonal group of indefinite quadratic forms in three dimensions. (I’ll give away the punch line right away: it’s because the parabola is a conic section.) This is not a particularly deep or significant observation, and will not be surprising to the experts, but I would like to record it anyway, as it allows me to review some useful bits and pieces of elementary linear algebra.

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