Is the symmetric closure of a relation which is already transitive, itself transitive?

Question

Let $R$ be a transitive relation. Is the symmetric closure of $R$ also transitive?

Answer 1

Counterexample, adapted from @amWhy's attempt based on an observation by @aefrrs: if $a\ne b\ne c\ne a$, the transitive relation $\{(a,\,b),\,(b,\,c),\,(a,\,c)\}$ has symmetric closure $\{(a,\,b),\,(b,\,c),\,(a,\,c),\,(b,\,a),\,(c,\,b),\,(c,\,a)\}$, which isn't transitive as it doesn't own $\{a,\,a\}$. As @NoahSchweber notes, a simpler example, whose transitivity is vacuous, is $\{(a,\,b)\}$ with $a\ne b$.