I need some help with technicalities concernig root-subspaces of nilpotent Lie algebras of operators.
Let $\mathfrak{g} \subseteq \mathfrak{gl}(V)$ be nilpotent Lie algebra, $\alpha \ \colon \ \mathfrak{g} \to K$ it's weight (i.e. linear functional, and there exists $v \in V$ such that $x v = \alpha(x) v$ for all $x \in \mathfrak g$).
Define $$V_{(\alpha)}(\mathfrak g) := \big\{ v \in V \ \colon \ \forall x \in \mathfrak g, \ \exists k \ge 1, \ \text{ such that } \ \big(x-\alpha(x) I\big)^k v = 0 \big\}.$$
Question 0: Is it true that $$V_{(\alpha)}(\mathfrak g) = \text{(increasing union)} \ \bigcup_{k \ge 1} V_{(\alpha)}^k(\mathfrak g)$$ where $$V_{(\alpha)}^k(\mathfrak g) := \big\{ v \in V \ \colon \ \forall x \in \mathfrak g, \ \big(x-\alpha(x) I\big)^k v = 0 \big\} = \bigcap_{x \in \mathfrak g} \operatorname{Ker}\big( (x-\alpha(x)I)^k \big) \ ?$$
Question 1: How can one prove that each $V_{(\alpha)}^k(\mathfrak g)$ is $\mathfrak g$-invariant?
Question 2: Does this hold: $V_{(\alpha)}^{k+1}(\mathfrak g)= \big\{ v \in V \ \colon \ \forall x \in \mathfrak g, \ (x-\alpha(x)I)v \in V_{(\alpha)}^k(\mathfrak g) \big\}$ ? (Inclusion $\supseteq$ is obviously true.)
I am very grateful for any help.