Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.
Let $\Phi$ be the root system of $(\mathfrak{g},\mathfrak{h})$, write $W$ for the corresponding Weyl group of $\Phi$, and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ corresponding to a root $\alpha$.
We fix a choice of positive roots $\Phi^+$, and let $\Delta$ be the corresponding subset of simple roots in $\Phi^+$. Note that each subset $I\subseteq\Delta$ generates a root system $\Phi_I\subseteq\Phi$, with positive roots $\Phi_I^+:=\Phi_I\cap \Phi^+$.
There are a number of subalgebras of $\mathfrak{g}$ associated with the root system $\Phi_I$. Let $ \mathfrak{l}_I:=\mathfrak{h}\oplus\sum_{\alpha\in\Phi_I}\mathfrak{g}_\alpha$ be the Levi subalgebra and let $ \mathfrak{u}_I:=\sum_{\alpha\in\Phi^+\backslash\Phi_I^+}\mathfrak{g}_\alpha $ be the nilpotent radical.
Then $\mathfrak{p}_I:=\mathfrak{l}_I\oplus \mathfrak{u}_I$ is a standard parabolic subalgebra of $\mathfrak{g}$.
We note that once $I$ is fixed, there is little use for other subsets of $\Delta$. Therefore, we omit the subscript if a subalgebra is obviously associated to $I$.
Definition 1: The category $\mathcal{O}^\mathfrak{p}$ is the full subcategory of $U(\mathfrak{g})$-Mod whose objects $M$ satisfy the following conditions:
$M$ is a finitely generated $U(\mathfrak{g})$-module.
$M$ is a direct sum of finite-dimensional simple $U(\mathfrak{l})$-modules.
$M$ is locally finite as a $U(\mathfrak{p})$-module.
Definition 2: The category $\mathcal{O}^\mathfrak{p}$ is the full subcategory of $U(\mathfrak{g})$-Mod whose objects $M$ satisfy the following conditions:
$M$ is a finitely generated $U(\mathfrak{g})$-module.
$M$ is a direct sum of finite-dimensional simple $U(\mathfrak{l})$-modules.
$M$ is locally finite as a $U(\mathfrak{u})$-module.
My question: How to show Definition 1 is equivalent to Definition 2?
