\(\mathrm{SO}(2,1)\) is the analog in two spatial dimensions of the Lorentz group \(\mathrm{SO}(3,1)\). Its generators \(\left(X_{1}, X_{2}, X_{3}\right)\) obey the Lie algebra \(\left[X_{i}, X_{j}\right]=\) \(c_{i j}^{k} X_{k}\) with

Use the metric tensor computed from Eq. (7.19) to show that \(\mathrm{SO}(2,1)\) is semisimple and non-compact.

Data from Eq. (7.19)

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[X1, X2] = X3 [X2, X3] = -X1 [X3, X] = X2.