The purpose of this study is to evaluate how well LY4213663 is tolerated and what side effects may occur in healthy participants and participants with rheumatoid arthritis (RA). The study drug will be administered either subcutaneously (SC) (under the skin) or intravenously (IV) (into a vein in the arm).

Blood tests will be performed to investigate how the body processes the study drug and how the study drug affects the body. The study will last approximately 33 weeks excluding screening.

Inclusion Criteria:

• Have a body mass index (BMI) of 18.0 to 32.0 kilogram per square meter (kg/m²), inclusive

Part C and D Only:

• Have a body weight at least 50 kilograms and BMI within the range 18.0 to 40.0 kg/m² (inclusive) at screening
• Have a diagnosis of adult-onset RA for at least 3 months prior to screening

Exclusion Criteria:

Healthy Participants for SAD Part A and MAD Part B Only:

• Have a history or presence of cardiovascular, respiratory, hepatic, ophthalmological, renal, gastrointestinal, endocrine, hematological, neurological, or psychiatric disorders capable of significantly altering the absorption, metabolism, or elimination of drugs

Participants with RA for Parts C and D Only:

• Have Class 4 RA according to American College of Rheumatology (ACR) revised criteria
• Have a 12-lead electrocardiogram (ECG) abnormality at screening
• Have a current or recent acute active infection