The transition from $\SO(7)$ to $\SO(8)$ is much the same as from $\SO(3)$ to $\SO(4)$: Flips generate $\SO(7)$, and two-sided multiplication generates the rotations with the real direction. To see the latter property, consider the transformation \begin{equation} x \longmapsto e^{\ell\alpha} x e^{\ell\alpha} \end{equation} We can divide $x\in\OO$ into a piece in the complex subalgebra containing $\ell$, and a piece orthogonal to this subalgebra. That is, since \begin{equation} \OO = \CC \oplus \CC^\perp \end{equation} we have \begin{equation} x = r e^{\ell\theta} + x^\perp \end{equation} where $x^\perp$ is orthogonal to $\ell$. Recall that imaginary octonions are orthogonal if they anticommute, that is \begin{equation} x \perp y \Longleftrightarrow \{x,y\}=xy+yx=0 \end{equation} so long as $\Re(x)=0=\Re(y)$, and that \begin{equation} e^{u\alpha}y = ye^{-u\alpha} \label{uconj} \end{equation} if $u\perp y$ (and again both $u$ and $y$ are imaginary). Thus, \begin{equation} e^{\ell\alpha} x e^{\ell\alpha} = e^{\ell\alpha} (r e^{\ell\theta} + x^\perp) e^{\ell\alpha} = r e^{\ell\theta+2\alpha} + x^\perp \end{equation} where we have used (\ref{uconj}).

Without further ado, we have \begin{align} \SO(8) &= \langle \{ x \longmapsto (u\cos\alpha+v\sin\alpha)(uxu)(u\cos\alpha+v\sin\alpha) : \nonumber\\ &\qquad u,v,x\in\OO, u^2=-1=v^2, \{u,v\}=0 \} \rangle \nonumber \\ &\qquad \hphantom{x}\cup \{ x \longmapsto pxp : p,x\in\OO, |p|=1 \}\rangle \end{align} However, both of these types of transformations are (generated by) symmetric multiplication. Furthermore, unlike with the quaternions, single-sided multiplication with the octonions actually generates all of $\SO(8)$, not merely a subset of it. This is an important property of $\SO(8)$, known as triality, which says that each of the following representations is equivalent: 1) \begin{align} \SO(8) &= \langle \{ x \longmapsto pxp : p,x\in\OO, |p|=1 \} \rangle \label{so8v}\\ \SO(8) &= \langle \{ x \longmapsto px : p,x\in\OO, |p|=1 \} \rangle \label{so8s}\\ \SO(8) &= \langle \{ x \longmapsto xp : p,x\in\OO, |p|=1 \} \rangle \label{so8d} \end{align} This difference between $\SO(8)$ and $\SO(4)$ is due to nesting. Single-sided multiplication generates $\SO(8)$; unlike over the quaternions, iterated multiplications do not collapse to a single operation over the octonions.