The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that
$$
\vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert X\vert)^{2-\vert \alpha\vert}.
$$
Assuming that $a\in\Sigma^1$ is real-valued principal type and denoting by $A$ its Weyl quantization, using the fact that $A$ is continuous on the Schwartz space
$\mathscr S(\mathbb R^n)$ and on its dual (tempered distributions)
$\mathscr S'(\mathbb R^n)$,
we may define the maximal extension $H$ of $A$ with the domain
$$
D(H)=\{u\in L^2(\mathbb R^n), Au \in L^2(\mathbb R^n) \}.
$$

Then I claim that $H$ is self-adjoint. I believe that it is well-known and I am looking for a reference in the literature.

A related question is the same problem for first-order pseudo-differential operators on a compact manifold without boundary $\mathcal M$ (equipped with a smooth density): let $A$ be a first-order pseudo-differential operator on $\mathcal M$ (I do not want to assume ellipticity, but I know that $A$ is continuous on $C^\infty(\mathcal M)$ and on the distributions on $\mathcal M$) and assume that $A$ is symmetric, that is such that for $\phi, \psi\in C^\infty(\mathcal M)$ $\langle A\phi,\psi \rangle=\langle \phi,A\psi \rangle_{L^2(\mathcal M)}$. Then consider the maximal extension $H$ of $A$ with $$ D(H)=\{u\in L^2(\mathcal M), Au \in L^2(\mathcal M) \}. $$ Then $H$ is self-adjoint. Is it true and well-known?

Last but not least, dropping the compactness assumption on $\mathcal M$ in the second question above, assuming that $A$ is properly supported, can I get the same result?