Exercise 12, § 9

Lie Groups and Lie Algebras, III, 399

Let $(e_1, e_2, e_3)$ be the canonical basis of $K^3$ . Consider the nilpotent Lie algebra structure on $K^3$ such that $[e_1, e_2] = e_3, [e_1, e_3] = [e_2, e_3] = 0$ . With the associated group law on $K^3$ (no. 5),

(x, y, z)(x', y', z') = (x + x', y + y', z + z' + \frac{1}{2}(xy' - yx')).

The mapping

(\lambda, \mu, v) \mapsto (\exp \lambda e_1)(\exp \mu e_2)(\exp v(e_1 + e_3))

of $K^3$ into G is neither surjective (show that $(0, 1, 1)$ is not in the image) nor injective (show that $(0, 1, 0)$ and $(1, 1, -1)$ have the same image).

Solution

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