It's going to be tough to work your core without initiating your lower back. Besides, the lower back is crucial to your core, so it should be trained just like every other muscle in your core.
Here is the first circuit I followed in PT after recovering from a parse fracture:
- 2 to 3 ab/lower back stretches to loosen up (e.g. back arches on all fours, cobra pose, child's pose, etc.).
- Regular plank
- Side planks
- Dead bugs / Alternating 6 inches
- Weighted squat
- Weighted forward lunge
- Weighted side lunge
- Standing knees to elbows (be sure to keep your abs tense when doing these)
- Hip lifts / Elevated hip lifts
This circuit isn't exhausting and leaves a lot of room for customization (rep count, hold lengths, weight, how many times you run the circuit, etc.). If you can hold a regular or side plank for over 60 seconds, I recommend you replace the regular and sides planks with thai planks and thai crucifixes; they do wonders and engage your core's stabilization muscles because they force you to balance. Mason twists are also very good at engaging most of your core, especially when weighted.
I don't recommend you do core exercises that are designed to put a lot of compressing strain on your lower back, like supermans, since you already have back issues.
If you want to really focus on your abs (since you say you pulled a lower ab muscle), you're going to have to engage your lower back a little. A simple circuit would be:
- In & Outs
- Bicycles (both forward and backward)
- Crunchy frogs
- Cross leg / wide leg situps
- V-up rollups
If you want more challenging ab exercises, check out the rest of the P90X's Ab Ripper X exercises; you can find the circuit free online. Warning, most of the other ab exercises in the circuit will put a lot of strain on your lower back.