# Generators Of $SO(n)$

I was wondering, for the group $$SO(n)$$, as far as I understand, the $$n\choose 2$$ infinitesimal rotations in the plane spanned by $$e_i$$ and $$e_j$$ for $$0\le i form a basis for the corresponding Lie algebra $$\mathfrak{so}(n)$$. But I wonder if there exists a smaller than $$n\choose 2$$ set $$\mathcal{X}\subseteq\mathfrak{so}(n)$$ such that $$\{\exp(tX):t\in\mathbf{R},X\in\mathcal{X}\}$$ generates the whole group?

• As a starting point, consider the fact that $g_1, g_2 \in \mathrm{SO}(n)$ are conjugate if and only if they rotate through the same oriented angle. So for example, $g_x^{}\,g_y^{}\,g_x^{-1} = g_z^{} \in \mathrm{SO}(3) \subseteq \mathrm{SO}(n)$, where $g_z^{}$ is the rotation by $\pi/2$ about the $z$-axis, etc. So we can certainly get away with fewer generators for $n \geq 3$. Commented Aug 1 at 14:43
• Thanks, that is really useful! But how would I get a smaller rotation around the $z$-axis? If I choose a smaller angle than $\pi/2$, this equation wouldn't work, right? Commented Aug 1 at 14:54
• This is where the smooth parameter $t$ comes in! If $g = \exp(X)$ for some $X \in \mathfrak{so}(n)$ describes a rotation about a certain axis $v$ through some angle $\theta$, then $\exp(tX)$ is a rotation about the same axis $v$ and in the same direction as $g$, but through an angle $t\theta$. Conjugation will send any small rotation to a different small rotation via the element of the group that maps one oriented axis to the other. Commented Aug 1 at 15:02
• Ah, sorry, now I understand! You conjugate a small rotation by a $\pi/2$ one. Cool, thanks a lot! Commented Aug 1 at 15:31
• Any two generic 1-parameter subgroups will generate $SO(n)$. Commented Aug 1 at 15:34