I consider a representation of a semi-simple Lie algebra $\mathfrak{g}$ (specifically, the symplectic and orthogonal Lie algebras $\mathfrak{sp}(2N)$ and $\mathfrak{so}(2N)$) as anti-Hermitian operators acting on some Hilbert space $\mathcal{H}$, i.e., $\rho(K)\rho(K')-\rho(K')\rho(K)=\rho([K,K'])$ and $\rho(K)^\dagger=-\rho(K)$ for $\rho: \mathfrak{g}\to \mathrm{Lin}(\mathcal{H})$.

My understanding is that if we now exponentiate $e^{\rho(K)}$ and take arbitrary products of these operators, we will get a unitary representation of a group, whose Lie algebra is $\mathfrak{g}$. Specifically, if $\mathfrak{g}$ is the symplectic algebra, the resulting group will be symplectic group or one of its $n$-fold covers (potentially even its universal cover with $n=\infty$). If $\mathfrak{g}$ is the orthogonal algebra $\mathfrak{so}(2N)$ with $N>1$, the resulting group would be either $SO(2N)$ or its double cover $Spin(2N)$, which is also its universal cover.

My questions: Are there requirements that I am missing to make this true? And under the conditions where this is true, do you know a standard reference that shows this or closely related statement?

Thank you so much.