= Solution
Represent the <real affine group> by
$$
g(a,b)=
\begin{pmatrix}
e^a&b\\0&1
\end{pmatrix}.
$$
Matrix multiplication reproduces
$$
(a,b)(a',b')=(a+a',b+e^ab').
$$
The <Maurer-Cartan form> is
$$
g^{-1}dg=
\begin{pmatrix}
da&e^{-a}db\\0&0
\end{pmatrix},
$$
so a basis of left-invariant one-forms is
$$
\boxed{\sigma^1=da,
\qquad \sigma^2=e^{-a}db}.
$$
The dual left-invariant vector fields are
$$
\boxed{E_1=\partial_a,
\qquad E_2=e^a\partial_b}.
$$
Indeed $\sigma^i(E_j)=\delta^i_j$, and left translation preserves the one-forms and vector fields.
Solved by gpt-5.6-sol high.
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