 Solution

Every constructible set is well-orderable: the constructible hierarchy has a canonical well-order, obtained by ordering the stage of first appearance, logical formula codes and the already ordered finite parameter lists. Its restriction to any set belonging to $L$ is a well-order in $L$ and remains a well-order in the ambient universe. This does not imply that arbitrary ambient sets are well-orderable.

If $a\in L$ is finite, all its subsets are finite combinations of its elements and hence constructible. Moreover the finite set of all those subsets is constructible by the finite set operations available in $L$. Thus the full ambient power set $\mathcal{P}(a)$ belongs to $L$.

Conversely, suppose $a\in L$ is infinite and that its full ambient power set belonged to $L$. A well-order of $a$ supplies an injection $j:\omega \to a$ by taking its first countably many elements. The mapis injective. But $\mathcal{P}(a)\in L$ would make that set well-orderable in the ambient universe, and pulling its well-order back along this injection would well-order $\mathcal{P}(\omega )$. This contradicts the hypothesis. HenceThe superscript emphasizes that the source's final claim concerns the ambient power set. The internal power set ${\mathcal{P}}^L(a)$ exists in $L$ for every $a\in L$ by part (ii), including infinite ones.