Let $L$ be an indecomposable nilpotent Lie algebra (finite dimensional and over $\mathbb{C}$). Is it possible for the last non-zero term of the central series to be strictly smaller than the center?
For context, as shown in The center of a nilpotent Lie algebra intersects each ideal any non-trivial ideal will intersect non-trivially with the center. But in If L is nilpotent then $K\cap L^n \not=0$ it is shown that it is possible to have a non-trivial ideal which intersects the last non-zero term of the central series trivially. However, the example given is decomposable, and the answerer himself wonders in a comment whether it is also possible with an indecomposable Lie algebra.
Also note that a Lie algebra has the property that any non-trivial ideal has non-trivial intersection with the last non-zero term of the central series iff that term coincides with the center.
Yes, it is possible. Here is an example of an indecomposable $7$-dimensional $5$-step nilpotent Lie algebra $L$, with basis $(x_1,\ldots ,x_7)$ and brackets $$ [x_1,x_2]=x_3,\,[x_1,x_3]=x_4,\,[x_1,x_4]=x_6,\,[x_1,x_6]=x_7,\,[x_2,x_3]=x_5, $$ where we have $L^1=L$, $L^2=\langle x_3,\ldots ,x_7\rangle $, $L^3=\langle x_4,\ldots ,x_7\rangle $, $L^4=\langle x_6,x_7\rangle $, $L^5=\langle x_7\rangle $, and $L^6=0$, but where the center has dimension $2$, namely $Z(L)=\langle x_5, x_7\rangle$.