Do Lie Groups need to satisfy the “addition property”? In these lecture notes I don’t understand why the step with
$$g(\omega)=(1+...)(1+...)...(1+...)$$
is justified. It would be justified if we had the following property:
$$g(\omega_1)g(\omega_2) = g(\omega_1 + \omega_2).$$
Is this a necessary assumption for a Lie group? Or is it a provable assumption? Or is the step justified for some other reason?

Edit: After reflection, I believe that this is not a necessary assumption; however, this is equivalent to choosing the "affine parameterization" - which always can be done.
 A: Going from the first equation (MacLaurin series expansion) to the second (general expansion, upon division of parameter space into N equal parts) is done under the following assumption:
$g$ is an element of a connected neighbourhood of the origin/identity $e\in G$ (called the connected component of the identity).
As of such, it is always a member of a 1-parameter subgroup $h(\omega)$ by the following theorem:

Theorem 2.9.3 (Cohn, P., "Lie Groups", CUP, 1965, p. 58)
Any connected 1-dimensional Lie group is analytically isomorphic either to the additive group of real numbers $\mathfrak{R}$, or to $\mathfrak{I}$, the additive group of real numbers mod 1. In particular, any 1-dimensional Lie group is locally isomorphic to $\mathfrak{R}$.

What does this theorem mean? That any $g\in G_0$ (the connected component of the identity) is expressible as a $h(\omega)$ with:
$$h(\omega_1)h(\omega_2) = h(\omega_1 + \omega_2) \tag{1}$$
Try to transform my $(1)$ into your second equation by considering the split of product.
