Here is the definition of a nilpotent Lie algebra given in Serre's book Lie Groups and Lie Algebras.
If $\mathfrak h$ is a Lie algebra, the center of $\mathfrak h$ is defined to be the set of $X \in \mathfrak h$ such that $[X,Y] = 0$ for all $Y \in \mathfrak h$. Then center of $\mathfrak g/\mathfrak a_{i+1}$ should be the set of $X + \mathfrak a_{i+1}$ such that $[X,Y] \in \mathfrak a_{i+1}$ for all $Y \in \mathfrak g$.
In that case, if $\mathfrak a_i/\mathfrak a_{i+1}$ is the center of $\mathfrak g/\mathfrak a_{i+1}$, then $[\mathfrak g, \mathfrak a_i] \subset \mathfrak a_{i+1}$. Serre indicates that the converse of this statement holds. But this seems obviously false. Am I missing something?
The phrase "is the center" should be "is contained in the center". That's all you need for a central extension.