The Interleaved Transition System for 2 Independent and Concurrent Transition Systems This question is about Model Checking for Software Formal Verification
How do you model the joint behavior of 2 independent and concurrent transition systems? 

Specifically, given the two independent and concurrent Transition Systems below; TS1 and TS2 (left to right).


A tutor proposed that the resulting Interleaving Transition System ITS is 


I understand how all the shown states and transitions of this ITS were gotten, however, why is there no transition from state (l1,q1) to (l3,q2)--as would be expected if both TS1 and TS2 transitioned on action a?
By definition, the interleaving operator is not synchronized, so any transition in $TS1\ |||\ TS2$ corresponds to a transition in a single component while the other one remains in whatever state it is.

More generally, in process algebras like CSP, the parallel composition operator $||$ can be parameterized with a set of symbols on which to synchronize - for example, you could consider $TS1\ |[\{a\}]|\ TS2$, which has the same states and $b$-transitions as $TS1\ |||\ TS2$, but only two (synchronized) $a$-transitions $(l_1,q_1)\to(l_3,q_2)$ and $(l_2,q_1)\to(l_1,q_2)$. In particular, $|||$ is the special case $|[\emptyset]|$.

The point is that (together with additional operations like hiding) this allows you to represent a far greater variety of models of interaction.