Comment by ajkjk

Nah it really is simpler than that, that picture has exploded the summations to make it look complicated. Although it is strangely hard to find the compressed version written down anywhere...

the thing about Lagrangians is that they compose systems by adding terms together: L_AB = L_A + L_B if A and B don't interact. Each field acts like an independent system, plus some interaction terms if the fields interact. So most of the time, e.g. on Wikipedia[0], people write down the terms in little groups. But still, note on the Wikipedia page that there are not that many terms in the Lagrangian section, due to the internal summations.

[0]: https://en.wikipedia.org/wiki/Mathematical_formulation_of_th...