In the first displayed equation on p. 376 of the book we have the equality
$$e^{2\pi\pi_l(F_1)} = e^{2\pi i L_3}=I$$
The reason for the second equality is that "the eigenvalues of $L_3$ are integers" which is shown previously in the chapter.
It is not clear to me how the first equality comes about. Here the $F_i$ are a basis of $so(3)$ given by
$$ F_1 = \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 &-1\\ 0 & 1 & 0 \end{bmatrix};\quad F_2 = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ -1 & 0 & 0 \end{bmatrix};\quad F_3 = \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}, $$ $\pi_l(\cdot)$ is a representation of $so(3)$ on a $2l+1$ dimentional space and $L_3=i\pi_l(F_3)$.
The most straightforward thing I can think about is that this is a typo and the second term should be $e^{-2\pi i L_1}$ where $L_1$ is defined analogously as $i\pi_l(F_1)$. Though $L_1$ is not defined or analysed in the chapter, we can then invoke symmetry to say that just like $L_3$ its eigenvalues too are integers and so the second equality holds.
But am I missing some other way in which the equality as given in the book is actually correct?