Calling A Second Domain Within an Initial Domain - Event Sourcing

Question

I am trying to create a serverless, event sourced architecture using AWS resources. In the image I have two separate domains.

The right domain is working fine and intended.

The left domain requires the logic which is inside the state machine of the right domain (displayed using the lambda pointing towards the statemachine on the right domain).

Using event sourcing and decoupling, I do not think the right approach would be to create a new state machine or directly invoke the state machine on the right side. However, I am struggling to find another way of trigger this nested domain logic the proper way.

One option is that I could call the API gateway method from within the lambda of the left domain, however that itself is not the most performant.

Your help is much appreciated.

architecture diagram