You are correct in understanding the FINRA rule that limits day trading to no more than four trades in five business days for pattern day traders, who must maintain a minimum equity of $25,000 in their trading accounts. This rule applies to both margin and cash accounts, and it is enforced by brokers to ensure regulatory compliance.
If you have a margin account with one broker and a cash account with another broker, you would still be subject to the FINRA rule. This means that even if you use algorithms or trading strategies that exceed the limit of four trades in five days, you would still be restricted by the rule and could face penalties if you violate it.
It is important to be aware of the regulations and policies set by FINRA and the individual brokers you use. If you plan to engage in day trading, make sure you understand the specific restrictions and requirements of each broker, as well as the tax implications of using multiple accounts.