  We prove some results about the theory of independence in $\mathrm{NSOP}_{3}$
theories that do not hold in $\mathrm{NSOP}_{4}$ theories. We generalize
Chernikov's work on simple and co-simple types in $\mathrm{NTP}_{2}$ theories
to types with $\mathrm{NSOP}_{1}$ induced structure in
$\mathrm{N}$-$\omega$-$\mathrm{DCTP}_{2}$ and $\mathrm{NSOP}_{3}$ theories, and
give an interpretation of our arguments and those of Chernikov in terms of the
characteristic sequences introduced by Malliaris. We then prove an extension of
the independence theorem to types in $\mathrm{NSOP}_{3}$ theories whose
internal structure is $\mathrm{NSOP}_{1}$. Additionally, we show that in
$\mathrm{NSOP}_{3}$ theories with symmetric Conant-independence, finitely
satisfiable types satisfy an independence theorem similar to one conjectured by
Simon for invariant types in $\mathrm{NTP}_{2}$ theories, and give
generalizations of this result to invariant and Kim-nonforking types.
