Stroke: The most common cause of sudden onset left arm and left leg weakness is a stroke, particularly if the weakness is unilateral. Strokes occur due to interruption of blood supply to the brain, either due to a blockage (ischemic stroke) or a rupture (hemorrhagic stroke) of blood vessels, leading to damage of brain tissue responsible for motor function.

Multiple Sclerosis (MS): MS is an autoimmune disease that affects the central nervous system, leading to demyelination, inflammation, and a wide range of neurological symptoms, including weakness. The symptoms can be unilateral and may involve the arm and leg on the same side.
Spinal Cord Injury or Compression: Injury or compression of the spinal cord, particularly in the cervical region, can cause weakness or paralysis of the limbs below the level of injury, including the arm and leg on the same side if the injury affects one side of the spinal cord more than the other.
Peripheral Neuropathy: Certain types of peripheral neuropathy, such as mononeuropathy multiplex, can cause weakness in specific limbs, although this is less common for causing simultaneous arm and leg weakness on the same side.

Brain Tumor: A tumor in the brain, especially in the motor cortex or the spinal cord, can cause progressive weakness. It's crucial to identify and treat brain tumors early to prevent further neurological deterioration.
Subdural or Epidural Hematoma: Bleeding in the brain, either between the brain and the skull (subdural) or between the skull and its outer covering (epidural), can lead to increased intracranial pressure and compression of brain tissue, causing focal neurological deficits like arm and leg weakness.
Cervical Spondylotic Myelopathy: This condition involves compression of the spinal cord in the neck due to wear and tear of the spine, leading to symptoms such as weakness, numbness, and clumsiness in the limbs.

Acute Disseminated Encephalomyelitis (ADEM): ADEM is a rare autoimmune disease characterized by a sudden, widespread attack of inflammation in the brain and spinal cord, which can lead to symptoms including weakness, among others.
Neurosarcoidosis: Sarcoidosis is a disease in which abnormal immune cells, called granulomas, form in various parts of the body, including the brain and spinal cord, potentially causing neurological symptoms like weakness.
Mitochondrial Myopathies: These are a group of disorders caused by dysfunctional mitochondria, the energy-producing structures within cells, which can lead to muscle weakness, among other symptoms.