I am reading through "State Estimation for Robotics" by Timothy Barfoot and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions: $$ C = (C_2C_1^T)^{\alpha}C_1, \alpha \in [0,1] $$
$$ C , C_1, C_2 \in SO(3), $$ That is, we have a rotation matrix C that is an interpolation of two rotation matrices $C_2$ and $C_1$, with the interpolation parameter $\alpha$. $$ C_{21} = \exp(\phi\text{^}) = C_2C_1^T, $$ and $$ C' , C_1', C_2' \in SO(3) $$ are perturbation rotation matrices for $C$, $C_1$, $C_2$ respectively with the left differences
$$ \delta\varphi = \ln(C'C^T)^{∨} $$ $$ \delta\phi_1 = \ln(C_1'C_1^T)^{∨} $$
$$ \delta\phi_2 = \ln(C_2'C_2^T)^{∨}. $$ In finding a relationship between $\delta\varphi$ and $\delta\phi_1$ , $\delta\phi_2$, the author states that \begin{align} \exp(\delta\varphi\text{^})C \\ &= (\exp(\delta\phi_2\text{^})C_2C_1^T\exp(-\delta\phi_1\text{^}))^{\alpha}\exp(\delta\phi_1\text{^})C_1 \\ &\approx \exp((\phi + \textbf{J}(\phi)^{-1}(\delta\phi_2 - C_{21}\delta\phi_1))\text{^})^\alpha\exp(\delta\phi_1\text{^})C_1 \\ &\approx \exp((\alpha\phi + \alpha\textbf{J}(\phi)^{-1}(\delta\phi_2 - C_{21}\delta\phi_1))\text{^})\exp(\delta\phi_1\text{^})C_1 \end{align} , which is straight forward to me.
However, what I don't understand is the following expression (continued): $$\approx \exp((\alpha\textbf{J}(\alpha\phi)\textbf{J}(\phi)^{-1}(\delta\phi_2-C_{21}\delta\phi_1))\text{^})C_{21}^{\alpha}$$
While trying to understand this, I was thinking I could understand it along the line of:
$$\delta\phi_3 = \delta\phi_2 - C_{21}\delta\phi_1$$ $$\phi_3 = \alpha\phi$$ and $$\alpha\textbf{J}^{-1}(\phi)$$ $$=\textbf{J}^{-1}(\alpha\phi)$$ $$=\textbf{J}^{-1}(\phi_3)$$ due to the linear operator property of Jacobian so that the last expression is expressive of something like $\exp(\phi_3 + J(\phi)^{-1}\delta\phi_3)\text{^}) \approx \exp(\delta\phi_3\text{^})\exp(\phi_3\text{^}))$, but I don't think it quite matches up with the final expression.. Also, I am not sure where the $\textbf{J}(\alpha\phi)$ is coming from.
Could someone please explain to me how this line is derived? This book is my first encounter with Lie Algebra.
Much thanks in advance.