The Dr. Navarro and Dr. Knabb addressed the presence of lighning in TS and hurricanes. Viewers probably got the impression that lightning is common in the tropics. Lightning is uncommon to rare in the tropics. It is only when tropical systems make their way farther north that they generate an increase in the rate of electrical activity.
Yes! There is sometimes lightning in tropical cyclones, but it is relatively rare. In fact there are some studies that suggest using lightning strikes as a good proxy for when strengthening is about to occur in these systems. They are rather lightning light because unlike T-storm complexes over land, the updrafts in the warm core eye wall area of a tropical system are relatively weaker. This prevents some of the ice formation needed to build up different charges within the storm and thus less lightning is found.