Yes, the first order approximation using the adjoint is correct.
It is easyer to see if one interpret members of the Lie algebra as minimal vectors $\mathbf{x}$ instead of square matrices $\widehat{\mathbf{x}}$. Thus, we define the Lie bracket as $[\mathbf{a},\mathbf{b}]:=\widehat{\mathbf{a}}\cdot\widehat{\mathbf{b}}-\widehat{\mathbf{b}}\cdot\widehat{\mathbf{a}}$.
In case of SO3, it is simply the cross product: $[\mathbf{a},\mathbf{b}]=\mathbf{a}\times\mathbf{b}$. However, a Lie bracket in such a vector from exists for all other matrix Lie groups too.
Accordingly, the BCH-Formular is now defined as $\text{bch}(\mathbf{a},\mathbf{b}):=\log(\exp(\widehat{\mathbf{a}})\exp(\widehat{\mathbf{b}}))^\vee$
For instance, as a third order approximation, we get:
$$\left. \frac{\partial}{\partial \mathbf{x}} \log(\mathtt{A}\exp(\mathbf{x})\mathtt{B})^\vee\right|_{\mathbf{x}=\mathbf{0}} $$
$$ =\left.\frac{\partial}{\partial \mathbf{x}} \log\left(\exp(\widehat{\mathtt{Ad}_\mathtt{A}\mathbf{x}})\mathtt{A}\mathtt{B}\right)^\vee\right|_{\mathbf{x}=\mathbf{0}}$$
$$=\left.\frac{\partial}{\partial \mathbf{x}} \log\left(\exp(\widehat{\mathtt{Ad}_\mathtt{A}\mathbf{x}})\exp(\widehat{\mathbf{c}})\right)^\vee\right|_{\mathbf{x}=\mathbf{0}}$$
$$= \left.\frac{\partial}{\partial \mathbf{x}} \text{bch}(\mathtt{Ad}_\mathtt{A}\mathbf{x},\mathbf{c})\right|_{\mathbf{x}=\mathbf{0}}$$
$$\approx \frac{\partial}{\partial \mathbf{x}}\left( \mathtt{Ad}_\mathtt{A}\mathbf{x}+\mathbf{c}
+ \frac{1}{2}[\mathtt{Ad}_\mathtt{A}\mathbf{x},\mathbf{c}]+\frac{1}{12}([\mathtt{Ad}_\mathtt{A}\mathbf{x},[\mathtt{Ad}_\mathtt{A}\mathbf{x},\mathbf{c}]]+ [\mathbf{c},[\mathbf{c},\mathtt{Ad}_\mathtt{A}\mathbf{x}]])\right)_{\mathbf{x}=\mathbf{0}}$$
$$ =\left.\left(\frac{\partial \mathbf{y}}{\partial\mathbf{y}} + \frac{1}{2}\cdot\frac{\partial [\mathbf{y},\mathbf{c}]}{\partial \mathbf{y}}\right|_{\mathbf{y}=\mathtt{Ad}_\mathtt{A}\mathbf{0}}
+\frac{1}{12}\left(\frac{\partial[\mathbf{y},[\mathbf{y},\mathbf{c}]}{\partial\mathbf{y}}-\frac{\partial[\mathbf{c},[\mathbf{y},\mathbf{c}]}{\partial \mathbf{y}}\right)_{\mathbf{y}=\mathtt{Ad}_\mathtt{A}\mathbf{0}}\right)
\left.\frac{\partial \mathtt{Ad}_\mathtt{A}\mathbf{x}}{\partial \mathbf{x}}\right|_{\mathbf{x}=\mathbf{0}}$$
$$= \left(\mathtt{I}
+ \left.\frac{1}{2}\cdot\frac{\partial [\mathbf{y},\mathbf{c}]}{\partial \mathbf{y}}\right|_{\mathbf{y}=\mathbf{0}} + \frac{1}{12}
\left(\frac{\partial [\mathbf{y},[\mathbf{0},\mathbf{c}]]}{\partial \mathbf{y}}+
\frac{\partial [\mathbf{0},[\mathbf{y},\mathbf{c}]]}{\partial \mathbf{y}}
-\left.\frac{\partial [\mathbf{c},\mathbf{w}]}{\partial \mathbf{w}}\right|_{\mathbf{w}=[\mathbf{0},\mathbf{c}]}
\left.\frac{\partial [\mathbf{y},\mathbf{c}]}{\partial \mathbf{y}}
\right|_{\mathbf{y}=\mathbf{0}}
\right)\right)
\mathtt{Ad}_\mathtt{A}\nonumber$$
$$= \left(\mathtt{I}
+ \left.\frac{1}{2}\cdot\frac{\partial [\mathbf{y},\mathbf{c}]}{\partial \mathbf{y}}\right|_{\mathbf{y}=\mathbf{0}} + \frac{1}{12}\left.\frac{\partial [\mathbf{w},\mathbf{c}]}{\partial \mathbf{w}}\right|_{\mathbf{w}=\mathbf{0}}
\left.\frac{\partial [\mathbf{y},\mathbf{c}]}{\partial \mathbf{c}}\right|_{\mathbf{y}=\mathbf{0}}
\right)\mathtt{Ad}_\mathtt{A}$$
$$= \left(\mathtt{I}
+ \left.\frac{1}{2}\cdot\frac{\partial [\mathbf{y},\mathbf{c}]}{\partial \mathbf{y}}\right|_{\mathbf{y}=\mathbf{0}} + \frac{1}{12}\left(
\left.\frac{\partial [\mathbf{y},\mathbf{c}]}{\partial \mathbf{c}}\right|_{\mathbf{y}=\mathbf{0}} \right)^2
\right)\mathtt{Ad}_\mathtt{A}$$
A unitary operator is a diagonalizable operator whose eigenvalues all have unit norm. If we switch into the eigenvector basis of U, we get a matrix like:
\begin{bmatrix}e^{ia}&0&0\\0&e^{ib}&0\\0&0&e^{ic}\\\end{bmatrix}
which is obviously the exponential of a diagonal hermitian matrix.
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