Exercise 1, § 9

Let $G = \mathbf{SO}(3, \mathbf{R})$. Let $\Delta_1, \Delta_2$ be two orthogonal lines in $\mathbf{R}^3$ and $H_i$ the subgroup of G consisting of the rotations about $\Delta_i$ ($i = 1, 2$). Then $[L(H_1), L(H_2)]$ is a 1-dimensional Lie subalgebra of G distinct from the Lie subalgebra tangent to $(H_1, H_2)$ at $e$.