A reasonably simple way to disentangle this is to start from the group. Surely a rotation by an angle $\theta$ about $\hat z$ would be represented by the real matrix \begin{align} R_z(\theta)&= \left(\begin{array}{ccc} \cos\theta & \sin\theta & 0 \\ -\sin\theta &\cos\theta &0 \\ 0&0&1\end{array}\right)\, \tag{1} \end{align} etc. Note that of course (1) is NOT a diagonal matrix with complex entries, but a real matrix which cannot be made diagonal without introducing complex numbers.

The generator of infinitesimal rotation (defined without the "i" as is traditional in physics) \begin{align} \hat {\mathbb{L}}_z=\frac{d}{d\theta}R_z\bigl\vert_{\theta=0} \end{align} would be the real antisymmetric matrix \begin{align} \hat {\mathbb{L}}_z = \left(\begin{array}{ccc} 0 & 1 & 0 \\ -1 &0 &0 \\ 0&0&0\end{array}\right)\, \tag{2} \end{align} and NOT hermitian.

You see how the physics convention would differ as the generators are defined with an $i$ in it: \begin{align} \hat {{L}}_z=-i\frac{d}{d\theta}R_z\bigl\vert_{\theta=0}\, . \end{align}

The factor of "$i$" is of course not an issue if you are dealing with matrices with complex entries, such as $SU(2)$.

In dealing with real form and complex extensions, the mathematics way of doing things is less confusing although not familiar to physics. The only math/phys. book who consistently follows the math convention is

Cornwell, J.F., 1984. Group theory in physics. 2 (1984). Acad. Press.

If you deal with compact groups, then one can complexify and decomplexify without second thoughts. If you are dealing with non-compact groups (v.g. Lorentz), then one has to be careful as representations that are irreducible under the reals may become reducible over the complex (v.g. Lorentz again: if you're not allowed to take the combo $K\pm iL$ then the adjoint is irreducible and does not break into $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$).

A reasonably simple way to disentangle this is to start from the group. Surely a rotation by an angle $\theta$ about $\hat z$ would be represented by the real matrix \begin{align} R_z(\theta)&= \left(\begin{array}{ccc} \cos\theta & \sin\theta & 0 \\ -\sin\theta &\cos\theta &0 \\ 0&0&1\end{array}\right)\, \tag{1} \end{align} etc. Note that of course (1) is NOT a diagonal matrix with complex entries, but a real matrix which cannot be made diagonal without introducing complex numbers.

The generator of infinitesimal rotation (defined without the "i" as is traditional in physics) \begin{align} \hat {\mathbb{L}}_z=\frac{d}{d\theta}R_z\bigl\vert_{\theta=0} \end{align} would be the real antisymmetric matrix \begin{align} \hat {\mathbb{L}}_z = \left(\begin{array}{ccc} 0 & 1 & 0 \\ -1 &0 &0 \\ 0&0&0\end{array}\right)\, \tag{2} \end{align} and NOT hermitian.

You see how the physics convention would differ as the generators are defined with an $i$ in it: \begin{align} \hat {{L}}_z=-i\frac{d}{d\theta}R_z\bigl\vert_{\theta=0}\, . \end{align}

The introduction complex numbers is required at some point because of the insistence on using diagonal operators. The eigenvectors of (2) are complex combination of the basis vectors $\hat{\boldsymbol{e}}_{x,y,z}$.

The factor of "$i$" is of course not an issue if you are dealing with matrices with complex entries, such as $SU(2)$.

In dealing with real form and complex extensions, the mathematics way of doing things is less confusing although not familiar to physics. The only math/phys. book I know who consistently follows the math convention is

Cornwell, J.F., 1984. Group theory in physics. 2 (1984). Acad. Press.

If you deal with compact groups, then one can complexify and decomplexify without second thoughts. If you are dealing with non-compact groups (v.g. Lorentz), then one has to be careful as representations that are irreducible under the reals may become reducible over the complex (v.g. Lorentz again: if you're not allowed to take the combo $K\pm iL$ then the adjoint is irreducible and does not break into $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$).