Spanning Sk(K(4))±
The dimensions of Sk(K(4)) are known by a general formula
from the article, The cusp structure of the paramodular groups for degree
two, J. Korean Math. Soc., 50(2):445-464, 2013, by Cris Poor and David S. Yuen.
We can run Jacobi restriction to a sufficient order to obtain
rigorous upper bounds on Sk(K(4))±.
When these two upper bounds add to the known dim Sk(K(4)),
which is what happens for the weights in question,
then the two upper bounds must in fact be the actual dimensions of
Sk(K(4))±.
Then the candidate expansions provided by Jacobi restriction are in fact
actual expansions of bases of Sk(K(4))±.