= Solution
For a finite-dimensional real <Lie algebra> $\mathfrak h$, the <Lie third theorem> asserts the existence of a connected <simply connected> <Lie group> whose <Lie algebra> is $\mathfrak h$. There are two complementary views of the construction.
Locally, the <Baker--Campbell--Hausdorff formula> defines multiplication on a sufficiently small neighbourhood of zero in $\mathfrak h$. Its leading terms are $X+Y+\tfrac12[X,Y]$, its identity is zero, and inversion is $X\mapsto-X$. The convergent BCH series in finite dimensions and the <Jacobi identity> provide the local associative law. Continuing and globalizing this local <group> gives its <simply connected> integration; the globalization is a substantive step, rather than a claim that one exponential chart gives all of the <group>.
A matrix construction makes that global step more concrete. The <Ado theorem> gives a faithful finite-dimensional representation
$$
\iota:\mathfrak h\hookrightarrow\mathfrak{gl}_N(\mathbb R).
$$
Consider the left-invariant distribution $D_A=A\iota(\mathfrak h)$ on $\operatorname{GL}_N(\mathbb R)$. It is involutive because $\iota(\mathfrak h)$ is closed under <commutators>. The <Frobenius theorem> integrates it to its connected leaf through the identity. This leaf carries the intrinsic immersed-submanifold structure of the connected <Lie subgroup> generated by the matrix exponentials $e^{\iota(X)}$. Its tangent <Lie algebra> is exactly $\iota(\mathfrak h)$; multiplication and inversion preserve the leaf and give its <Lie group> operations. Call this immersed <group> $H_0$.
Taking its <universal covering Lie group> gives
$$
\boxed{\operatorname{Lie}(\widetilde H_0)\cong\mathfrak h,\qquad
\widetilde H_0\text{ connected and simply connected}.}
$$
Ado's faithfulness matters: the adjoint representation alone loses central elements. Also, the <subgroup> generated by the exponentials need not be closed in the ambient matrix <group>. For an irrational $a$, the map $t\mapsto(R_t,R_{at})$ gives a dense one-dimensional immersed <subgroup> of the two-dimensional torus. Its intrinsic <group> is $\mathbb R$, whereas its closure has a two-dimensional <Lie algebra>. Taking an ambient closure in the construction would therefore be incorrect.
Finally, any other connected <Lie group> with <Lie algebra> $\mathfrak h$ is obtained from this <simply connected> integration by a discrete central quotient, as in part (i). The <simply connected> integration is unique up to isomorphism, but an arbitrary corresponding <group> is not.
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