= Solution
For independence, construct four <first-order structures>, each satisfying exactly the three required remaining <axioms> and failing the chosen one.
To fail <associativity>, use the carrier $\{0,1,2\}$, interpret $c$ as $0$, and define the commutative operation by its table:
$$
\begin{array}{c|ccc}
F&0&1&2\\\hline
0&0&1&2\\
1&1&0&1\\
2&2&1&0
\end{array}
$$
The first row and column give a two-sided identity, and every element is its own inverse because the diagonal entries are $0$. <Symmetry> gives <commutativity>. But
$$
F(F(1,1),2)=2\ne0=F(1,F(1,2)),
$$
so <associativity> fails.
To fail the identity axiom, take $\{0,1\}$, $c=0$, and the constant operation $F(x,y)=0$. It is associative and commutative, and every element has a two-sided partner producing $c$; but $F(1,c)=0\ne1$.
To fail the inverse axiom, use <natural numbers> under <addition> with $c=0$. <Addition> is associative and commutative and has the required identity, but $1$ has no additive inverse in this carrier.
To fail <commutativity>, use the <symmetric group> $S_3$ under <permutation> composition with $c$ its identity. <Associativity>, identity and inverses are <group> properties, while $(12)(23)\ne(23)(12)$. Each countermodel witnesses the failure of <semantic entailment> from the other three to the failed <first-order sentence>. Hence \b[the four source <axioms> are independent].
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