$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

Let $\boldsymbol{\theta}$ represent a spatial rotational increment, such that a rotation tensor compounded by an infinitesimal incremental rotation is given by $$\begin{equation} \boldsymbol{\Lambda}_{\varepsilon}=\cay\left[\varepsilon\hat{\boldsymbol{\theta}}\right]\cdot\boldsymbol{\Lambda}\,.\label{eq:rbr-spatial-rotation-increment} \end{equation}$$ Using the Cayley transform for illustration, the linearization of $\boldsymbol{\Lambda}$ along the increment $\boldsymbol{\theta}$ is obtained from $$\begin{equation} \begin{aligned}D\boldsymbol{\Lambda}\left[\boldsymbol{\theta}\right] & =\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\cay\left[\varepsilon\hat{\boldsymbol{\theta}}\right]\cdot\boldsymbol{\Lambda}\\ & =\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\left(\mathbf{I}+\frac{2}{1+\frac{1}{4}\varepsilon^{2}\theta^{2}}\left(\frac{1}{2}\varepsilon\hat{\boldsymbol{\theta}}+\frac{1}{4}\varepsilon^{2}\hat{\boldsymbol{\theta}}^{2}\right)\right)\cdot\boldsymbol{\Lambda}\\ & =\hat{\boldsymbol{\theta}}\cdot\boldsymbol{\Lambda} \end{aligned} \,.\label{eq:rbr-spatial-rotation-linearization} \end{equation}$$ The same result may be obtained with the exponential map. Similarly, using an infinitesimal material rotational increment such that $\boldsymbol{\Lambda}_{\varepsilon}=\boldsymbol{\Lambda}\cdot\cay\left[\varepsilon\hat{\boldsymbol{\Theta}}\right]$ , we may find $$\begin{equation} D\boldsymbol{\Lambda}\left[\boldsymbol{\Theta}\right]=\boldsymbol{\Lambda}\cdot\hat{\boldsymbol{\Theta}}\,.\label{eq:rbr-material-rotation-increment} \end{equation}$$