I am following Classical and Quantum Mechanics via Lie Algebras by Neumaier and Westra.

Setup

I am stuck at part of Thm 2.3.1. Consider the matrix group $mathbb{G}$. The set of $mathbb{G}$-motions is the set of maps $U:[0,1]rightarrow mathbb{G}$ with $U(0) = I$. if $U$ is a $mathbb{G}$-motion then

begin{align}
f = frac{d}{dt}U(t)Bigrvert_{t=0}
end{align}

is called an infinitesimal motion. For example, if $U = e^{ft}$ then $frac{d}{dt}U(t)Bigrvert_{t=0} = f e^{ft}Bigrvert_{t=0} = f$ so $f$ is an infinitesimal motion. The authors denote the set of all infinitesimal motions by $mathbb{L} = log mathbb{G}$. The theorem in question state that

1) $mathbb{L}$ is a vector space and that if $f,g in mathbb{L}$ then $[f,g]=fg-gf in mathbb{L}$,

2) for $finmathbb{L}$ the adjoint mapping of $Vinmathbb{G}$ on $f$: $text{Ad}_V(f) = VfV^{-1} in mathbb{L}$

3) $text{Ad}_V [f,g] = left[text{Ad}_V f, text{Ad}_V g right]$

I understand points 2) and 3) and that $mathbb{L}$ is a vector space. What I want help with is the proof that $[f,g] in mathbb{L}$. I am trying to work through this proof without relying on the Landau notation $O(t^2)$ at any point.

Example of the type of proof I’m looking for

For example, it can be proven that $mathbb{L}$ is a vector space as follows: $f,g in mathbb{L}$, $alpha, beta in mathbb{K}$, the field. Since $f,g in mathbb{L}$ there exists $V_f(t)$ and $V_g(t)$ with $frac{d}{dt}V_f(t)Bigrvert_{t=0} = f$ and $frac{d}{dt}V_g(t)Bigrvert_{t=0} = g$. Consider then

begin{align}
V_{alpha,beta}(t) = V_f(alpha t)V_g(beta t)
end{align}

Then

begin{align}
frac{d}{dt}V_{alpha,beta}(t) &= left(frac{d}{dt}V_f(alpha t)right)V_g(beta t)Bigrvert_{t=0} + V_f(alpha t)left(frac{d}{dt}V_g(beta t)right)Bigrvert_{t=0}
&= alpha f I + I beta g = alpha f + beta g
end{align}

so $alpha f + beta g in mathbb{L}$ so $mathbb{L}$ is closed under addition and scalar multiplication. This proof didn’t rely on the big-O notation at all.

The proof I’m having trouble with and my questions

The proof in the document that $[f,g] in mathbb{L}$ is as follows. From part 2) and the first part of part 1) we have that $V_g(t)fV_g(t)^{-1} – f in mathbb{L}$. It can be seen that $frac{d}{dt}V_g(t)^{-1}Bigrvert_{t=0} = -g$ So then

begin{align}
V_g(t)fV_g(t)^{-1} – f = (1+gt+O(t^2))f(1-gt+O(t^2))-f = t[f,g] + O(t^2)
end{align}

Then divide by $t$ and take $trightarrow 0$ to see that $[f,g] in mathbb{L}$.

I have two problems with this proof.

A) it relies on the big $O(t^2)$ notation. I would prefer a proof that didn’t rely on that.

B) I can see that $frac{V_g(t)fV_g(t)^{-1}-f}{t} in mathbb{L}$ but it doesn’t seem obvious to me that the limit of this operator as $trightarrow 0$ should also be an element of $mathbb{L}$.

Could anyone please help me find a proof which doesn’t rely on big-O notation and also help me with this limit question?