respectively - where to place it

There is no way of judging whether the word is even correct in the sentence without seeing the inclusion of the 2 cross-section expressions.

The whole sentence is:
The cross-sections of the reactions X and Y turned out to be 2 and 3 times lower, respectively, than those of X element in the Y reaction.

snuppelinaThe whole sentence is:The cross-sections of the reactions X and Y turned out to be 2 and 3 times lower, respectively, than those of X element in the Y reaction.

Can a "cross-section" be "lower"?

Small would be better, I think

snuppelinaThe cross-sections of the reactions X and Y turned out to be 2 and 3 times lower, respectively, than those of X element in the Y reaction.

Frankly, I don't see how a cross-section of anything can be either 'lower' or 'smaller', but as far as the structure is concerned, there is no need for the commas:

The cross-sections of the reactions X

Thank you very much!
I thought 'respectively' always requires a comma. Would a comma be a mistake? Or is it simply a difference between American and British English?

Ah, I think I can feel the difference!

snuppelinaSmall would be better, I think

If a cross-section is said to be "smaller", I would understand that to mean "smaller in area". I don't know whether that makes sense in your case; I'm not sure in what sense you are using the term "cross-section".

snuppelinaI thought 'respectively' always requires a comma. Would a comma be a mistake? Or is it simply a difference between American and British English?

The commas with short adverbials are gradually being done away with; you will probably meet proponents of both viewpoints.