Solutions to the exercises
Solutions for Chapter 2 exercises
Solution to exercises 2.3.1 We start from the Jacobi identity and substitute the given formulas, using the bilinearity and skew-symmetry of the Lie bracket to simplify the results:
Since x1, x2, x3 are linearly independent, we can conclude that
which forces a = 1, b = −1.
If you think about it further, this calculation suffices to ensure the existence of the Lie algebra described in the question with these values of a and b, namely with basis x1, x2, x3 whose Lie bracket satisfies ...