C680 Captain reported the undocumented transported of a 500W portable lithium battery which exceeded the company Hazmat maximum of 160W. The flight crew was not notified of the 500W battery.

Narrative

On Day 0 We flew a family from ZZZ to ZZZ1. Apparently we unknowingly carried a 500W portable lithium battery in the baggage compartment. This is well in excess of the max 160W batteries that we can carry as stated in our Hazmat policy. The passengers had a large amount of baggage that nearly filled the CE-680 baggage compartment and stated that they were going camping. This alerted me to be on the lookout and ask about camping fuel; charcoal; stoves; etc. My partner loaded the bags with the help of two lineman while I greeted and briefed the passengers. Nothing was said about a large separate battery and we completed the flight uneventfully. A few days later another crew member notified me that they had picked up the passengers on their return flight from ZZZ1 and identified and refused carriage of the battery. The passengers had told them they had brought it with them. Of course that was only because I did not realize we had it on board. Ultimately it was my responsibility as PIC to verify what we were carrying on board and I neglected to verify with the passengers that we did not have a large spare battery in the baggage compartment.I must always be vigilant of what we are loading on our aircraft. Larger batteries are becoming more and more common in all types of items that passengers will want to carry onboard. They are much easier to spot when attached to a scooter; skateboard; or even large smart' bag. When asking passengers about spare batteries it's always been in the context of spare laptop or cell phone chargers. Until this incident it had not occurred to me that someone would bring along a large battery to power items while going camping. This is something that I will certainly be on the lookout for in the future."