Why do we prefer to use $i$ with generators in Lie Algebra

Question

I am reading A. Zee. Group theory in a Nutshell for Physicists and for some reason, he prefers to write the generators with an $i$ near them

For example, a rotation can simply be described as:

$$e^{\theta \mathcal{J}}$$ where:

$$\mathcal{J}= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$

but instead, we define $J=-i\mathcal{J}$ and write rotations as $e^{i\theta J}$.

Also, the generator for a boost is:

$$iK = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ so a boost is $e^{i\varphi K}$.

Also, we approximate a unitary operator as:

$$U(\epsilon) \approx I - i\epsilon \hat{H} \implies U(t) = e^{i\hat{H}t}$$

even though the $i$ is arbitrary. So why is that?

Edit:

So for the Time Evolution Operator part, this question answers it (which is related to Hermitian Operators), but why should this be the case for rotations and boost? Is it because of quantum mechanics again? Why should the generator of rotations be hermitian?

Because then the generator is hermitian, which e.g. is desirable in quantum mechanics.. This might be of interest. — Tobias Fünke, Jul 24 '22 at 18:18
I understand that for the unitary operator, but for the boost and rotation, why? — Habouz, Jul 24 '22 at 18:29
In quantum mechanics we 'implement' symmetries as unitary (or antiunitary) operators, cf. Wigner's theorem. — Tobias Fünke, Jul 24 '22 at 18:31
Does this answer your question? Lie Algebra Conventions: Hermitian vs. anti-Hermitian — ZeroTheHero, Jul 24 '22 at 18:31
@ZeroTheHero It answers only the time evolution operator part. But I don't understand why should this be the case for rotations and boosts. — Habouz, Jul 24 '22 at 18:33
@Habouz, in physics complex $i$ basically means phase of $90^{\circ}$. Also see here: https://physics.stackexchange.com/q/428223/ — Brendan Darrer, Jul 24 '22 at 18:36
Mathematicians don't bother with the i, for simplicity. In classical mechanics, rotations don't need the i. Some groups use it, some don't. Understand both. — Cosmas Zachos, Jul 24 '22 at 18:50

Why do we prefer to use $i$ with generators in Lie Algebra

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