Theorem. Given $H_1, \ldots, H_k\in\mathbb C^{d\times d}$ Hermitian let $\mathfrak{g} := \langle iH_1, ... iH_k \rangle_{\text{Lie}}$ denote the associated Lie algebra and let $e^{\mathfrak g}$ be the corresponding Lie group. Moreover, let $G \subseteq U(d)$ be the group generated by all individual one-parameter groups $\{e^{itH_j}:t\in\mathbb R\}$. Then $G=e^{\mathfrak g}$.
Proof. $e^{\mathfrak g}\subseteq G$: Note that $e^{\mathfrak g}$ is a connected Lie group so the inclusion in question follows from Lemma 6.2 in Jurdjevic, Sussmann, "Control Systems on Lie Groups", Journal of Differential Equations 12, p.313-329 (1972).
As a side note, in the qubit case such decompositions are known as Z-Y decomposition (and its generalizations), refer to Theorem 4.1 ff. in Nielsen & Chuang.
$G\subseteq e^{\mathfrak g}$: Given any $x\in G$, by definition of $G$ there exist $m\in\mathbb N$ and real numbers $\{t_{ab}:a=1,\ldots,k, b=1,\ldots,m\}$ such that $x=\prod_{j=1}^m e^{it_{1j}H_1}\cdot\ldots\cdot e^{it_{kj}H_k}$. Re-writing
$$
x=\prod_{j=1}^m e^{i|t_{1j}|{\rm sgn}(t_{1j})H_1}\cdot\ldots\cdot e^{i|t_{kj}|{\rm sgn}(t_{kj})H_k}
$$
shows that $x$ lies in the reachable set of the bilinear control system $\dot x(t)=\sum_{j=1}^ku_j(t)H_jx(t)$, $x(0)={\bf1}$ where the $u_j$ are piecewise constant functions which only take values $1$ and $- 1$. But this reachable set is equal to $e^{\mathfrak g}$, cf. Theorem 5.1 ff. in the previously cited Jurdjevic & Sussmann paper; hence $x\in e^{\mathfrak g}$ which concludes the proof. $\square$
Finally let me stress that your first inclusion $e^{\mathfrak g}\subseteq G$ does not follow from the arguments made in the paper of Lloyd you cited. The reason for this is that he implicitly requires either $e^{\mathfrak g}$ or $G$ to be closed (which need not be true in general).
- His first argument---the Lie product formula for commutators (Eq.(2) in his paper)---implies $e^{\mathfrak g}\subseteq\overline{G}$, where the closure $\overline{G}$ of $G$ comes into play because we have to allow for approximations = the limit $n\to\infty$. If $G$ is closed, then this does indeed show $e^{\mathfrak g}\subseteq G$. However, $G$ need not be closed: he common counterexample here are irrational windings on a torus, e.g., $H={\rm diag}(1,\sqrt2)$ in which case $$
G=\{e^{itH}:t\in\mathbb R\}\subsetneq\overline{\{e^{itH}:t\in\mathbb R\}}=\{{\rm diag}(e^{i\theta_1},e^{i\theta_2}),\theta_1,\theta_2\in\mathbb R\}\,,
$$
refer also to Proposition 2.5 in Elliott's book "Bilinear Control Systems", Springer (2009).
- In his second argument---the "noninfinitesimal" construction---Lloyd argues that "if $\mathfrak g$ has finite dimension, [the space $e^{\mathfrak g}$] is
compact: As a result, at most, a number of transformations
proportional to the number of generators of $\mathfrak g$ is required
to reach any desired transformation in the space".
The mistake made here is that $\dim\mathfrak g<\infty$ does in general not imply compactness of $e^{\mathfrak g}$; again, irrational windings on a torus show that $e^{\mathfrak g}$ need not be closed.
