  Loops and cycles play an important role in computing endomorphism rings of
supersingular elliptic curves and related cryptosystems. For a supersingular
elliptic curve $E$ defined over $\mathbb{F}_{p^2}$, if an imaginary quadratic
order $O$ can be embedded in $\text{End}(E)$ and a prime $L$ splits into two
principal ideals in $O$, we construct loops or cycles in the supersingular
$L$-isogeny graph at the vertices which are next to $j(E)$ in the supersingular
$\ell$-isogeny graph where $\ell$ is a prime different from $L$. Next, we
discuss the lengths of these cycles especially for $j(E)=1728$ and $0$.
Finally, we also determine an upper bound on primes $p$ for which there are
unexpected $2$-cycles if $\ell$ doesn't split in $O$.
