Put the pigeon peas, bay leaves, and garlic in a large pot, cover with 3 quarts of cold water, and place over medium heat. Cover and cook the beans until tender, about 1 1/2 hours. Check the water periodically; add more, if necessary, to keep the peas covered. Drain the pigeon peas and reserve 4 cups of the cooking liquid.
Preheat the oven to 375 degrees F.
Coat a large Dutch oven or other ovenproof pot with the achiote oil and place over medium heat. When the oil begins to smoke, add the onion, minced garlic, and peppers. Cook, stirring, for 10 minutes, until the vegetables have softened, without letting them brown. Stir in the cumin, coriander, and cayenne. Mix in the rice and stir. Add the reserved pigeon peas cooking liquid and stir. Add the peas, salt and lime juice. Top with a lid and place into the oven. Continue to cook until the liquid is evaporated, about 20 to 30 minutes, until the rice is tender and the liquid is absorbed.
To make the achiote oil, pour the oil and achiote seeds in a pot or skillet. Cook over medium-low heat for 5 to 10 minutes; the oil will be reddish-orange from the achiote. Strain the oil and discard the achiote seeds.