Liability insurance is built for accidents that hurt other people. Three common exclusions follow from that idea.
Intentional acts. California Insurance Code §533 says an insurer is not liable for a loss caused by the wilful act of the insured, but it is not excused by the insured's negligence. Homeowners policies carry a matching exclusion for injury the insured expected or intended. A deliberate punch is not covered, even after a court enters a judgment; a careless accident can be.
Business activities. Section II is meant for personal exposures, so injury arising out of the insured's business is generally excluded, even when it happens at the home. A client hurt at a home consulting office is a business exposure that needs business coverage.
Damage to the insured's own property. Liability pays for harm to others, so damage to property owned by an insured is excluded under Section II. The family looks to the property coverages of the policy (Section I) for its own losses.