It is known that the six of fundamentals in a simple Lie group SU(3) makes the following multiplication decompositions of simple Lie group Representations:
$$
3^6= 5({1})+{28}+16({8})+10({10})+...
$$
How do we know the expressions of 5 independent set of singlet $1$ in terms of six of fundamentals:
$$
g_a, g_b, g_c, g_d, g_e, g_f?
$$
For example, one object is
$$
\epsilon_{abc} g_a g_b g_c \epsilon_{def} g_d g_e g_f,
$$
The following does not work
$${\epsilon_{abcdef} g_a g_b g_c g_d g_e g_f.}\text{FALSE!}$$
because we have indices $a$,..., $f$ running in $1,2,3$, because it is fundamental.
What are all the 5 objects? How do we know whether they are 5 independent ones?