As an AI language model, I do not have the capability to take exams, including the English CET-4 and CET-6. However, I can certainly provide you with information on how to check your eligibility for the exams.
The English CET (College English Test) is a standardized English proficiency test conducted in China. It includes two levels: CET-4 and CET-6. These tests are usually taken by college students or graduates who want to demonstrate their English language skills for academic or employment purposes.

To check your eligibility for the English CET-4 and CET-6 exams, you can follow these steps:

1. Visit the official website: Go to the official website of the National Education Examinations Authority (NEEA) or the Ministry of Education in China. These websites contain all the necessary information regarding the English CET exams, including the registration process, eligibility requirements, and exam dates.
2. Read the eligibility criteria: Look for the specific requirements for CET-4 and CET-6 exam registration. Typically, CET-4 is open to third-year undergraduates or above, while CET-6 is available to fourth-year undergraduates or above. Ensure that you meet the academic year criteria mentioned on the website.
3. Check academic prerequisites: Verify if your university or college is recognized by NEEA for CET registration. In some cases, only students from certain institutions are allowed to take the exams. Make sure your institution is listed as eligible.
4. Gather necessary documents: Collect all the required documents, such as your student ID card, passport, and recent photographs as per the guidelines provided on the official website. These documents may be needed during the registration process.
5. Registration process: once you have confirmed your eligibility and gathered the necessary documents, proceed with the online registration process. Follow the instructions carefully and provide accurate information. Pay attention to registration deadlines and fees, as they vary each year.
6. Confirmation and exam details: After completing the registration, you will receive a confirmation email or notification containing your exam details, including the date, time, and venue. Make sure to save this information for future reference.
Remember, it is essential to check the official website regularly for updates or changes regarding the English CET exams. Sometimes, there might be modifications in the eligibility criteria or exam format.

By following these steps and conducting proper research, you can determine your eligibility and proceed with the registration process for the English CET-4 and CET-6 exams. Good luck with your preparations, and I hope you achieve successful results!