Help in ISO 17025 implementation

You asked

Will a reference standard be accepted that will expire in June 2023, knowing that the laboratory seeks effective application of ISO 17025?

As long as the certified reference material (CRM)  has been stored according to requirements, and it is used before expiry, you can show acceptability to use it and traceability of your results through the material's certified value.

It is however always best practice to test one lot number of a reference material against an existing material for which you have quality control data. This will provide reassurance in the certified value, particularly if you are close to the expiry date.

You also asked

if it is mentioned in the certificate of the reference material that the re-test of the material in 2024, what is the method used to re-test and verify the effectiveness and stability of the material? The test result"

The specific test depends on the material and method technique. You need to show that the certified property value/s are not significantly changed. The depth of the assessment depends on the expected influence of the CRM on the validity of the measurement and the criticality of the measurement. Use statically tools to determine if there is any significant difference in the measured property of the CRM compared to the quality control data or validation data of the same CRM when first introduced I,e before expiry.

As stability is being evaluated, and there is a possibility of breakdown components, the test must be specific and selective. Typically you will run a number of replicates of the CRM as a test sample using your normal process and quality control checks.