Why are there only two linearly independet quartic Higgs terms for the adjoint $24$ in $SU(5)$ GUTs?

Question

Given the adjoint $24$ of $SU(5)$, we have

$$ 24\otimes 24 = 1_s \oplus 24_s \oplus 24_a \oplus 75_s \oplus 126_a \oplus \overline{126_a} \oplus 200_s ,$$

where each representation is denoted by its dimension and the subscripts $s$ and $a$ denote symmetric and antisymmetric respectively. Naively, I would say we have 7 quartic invariants:

$$ (24\otimes 24)_{1_s} (24\otimes 24)_{1_s} + (24\otimes 24)_{24_s} (24\otimes 24)_{24_s} + (24\otimes 24)_{24_a} (24\otimes 24)_{24_a} + (24\otimes 24)_{75_s} (24\otimes 24)_{75_s} + (24\otimes 24)_{126_a} (24\otimes 24)_{126_a} + (24\otimes 24)_{\overline{126_a}} (24\otimes 24)_{\overline{126_a}} +(24\otimes 24)_{200_s} (24\otimes 24)_{200_s} ,$$

because

$$ 1_s \otimes 1_s = 1 \quad 24_s \otimes 24_s =1 \quad 75_s \otimes 75_s =1 \quad etc. $$

Nevertheless, in all $SU(5)$ papers only two quartic terms appear in the $24$ Higgs potential. How can I compute how many and which of these 7 terms are linearly independent?

This post imported from StackExchange Physics at 2015-10-05 20:44 (UTC), posted by SE-user JakobH