Here Lie algebras/groups are real. The most straightforward definition might be:

Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian form $\langle \cdot, \cdot \rangle$ such that $\langle \rho(g)v,w \rangle = -\langle v, \rho(g)w\rangle$ for all $v,w \in V$.

But it seems like there should be a definition in which, when one has a (strongly continuous) unitary representation of a Lie group $G$, it yields a unitary representation of its Lie algebra $\mathfrak{g}$ via differentiation. So we should allow for the elements of $\mathfrak{g}$ to act as possibly unbounded skew adjoint operators. So one might say:

Def(?): A unitary representation of a Lie algebra $\mathfrak{g}$ on a Hilbert space $\mathcal{H}$ is a linear map $\rho:\mathfrak{g} \rightarrow \{\text{Possibly unbounded skew adjoint operators on $\mathcal{H}$}\}$, such that $[\rho(g),\rho(h)]$ is essentially skew adjoint, and its unique skew adjoint extension equals $\rho([g,h])$.

So my question is: Does this second definition capture what I want (In that, say, there is a one-to-one correspondence between unitary reps of $G$ and $\mathfrak{g}$ when $G$ is simply connected), and is there any nice relationship between the objects satisfying definition (1) and the objects satisfying definition (2)? (For instance, for a vector space satisfying the first definition one can take its completion- and this is possibly an object satisfying the second definition?)


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