If $T$ is a triangulated category, then the formalism of $t$-structures gives a way to find abelian subcategories inside. You're supposed to find two strictly full subcategories, $T^{\le 0}, T^{\ge 0}$ so that

1) $T^{\ge 0}[-1] \subset T^{\ge 0}$ and $T^{\le 0}[1] \subset T^{\le 0}$

2) There are no nonzero maps from an object in $T^{\le 0}[1]$ to an object in $T^{\ge 0}$

3) For any object $K$ there's a triangle $A \to K \to B$ with $A \in T^{\le 0}[1]$ and $B \in T^{\ge 0}$.

But if $T$ is periodic, say $[n]$ is the identity, this does not seem like a recipe for success.

How do you find abelian subcategories of a periodic triangulated category?