I have lost my product key for office 365 and can't install outlook classic on my new computer.

You do not need the lost product key to reinstall Microsoft 365 or Outlook classic; sign in with the Microsoft account originally used to purchase or redeem the subscription.

Install and activate Outlook classic

1. Open the Microsoft account Services & subscriptions page:
   https://account.microsoft.com/services/
2. Sign in with the Microsoft account used to buy or activate Microsoft 365.
3. Find the active Microsoft 365 subscription and select its installation option.
4. Run the downloaded installer.
5. Open Outlook and, if activation is requested, sign in with that same Microsoft account instead of entering a product key. Microsoft states that the account replaces the key for reinstalling and activating Microsoft 365.^1^

If the Microsoft 365 installation gives you only new Outlook, install classic Outlook separately:

1. Open Microsoft’s classic Outlook installation page.^2^
2. Under Select a language and download classic Outlook, choose your language.
3. Select Download.
4. Run the downloaded setup file and complete the prompts.
5. Start Outlook (classic) from the Windows Start menu, sign in for licensing if prompted, and then add your email account.

Many new Windows computers include new Outlook but not classic Outlook, so classic Outlook may require this separate installation.^2^

If the subscription is not displayed

Try any other Microsoft account that may have been used for the original purchase. On the old computer, if available, open Word or another Office application and select File → Account; the address shown after Belongs to identifies the associated account.^1^

If no account shows the subscription and the key was never redeemed, Microsoft Support may be able to replace a lost key or provide another option when you have valid proof of purchase.^1^

References

1. Using product keys with Microsoft 365 | Microsoft Support
2. Install or reinstall classic Outlook on a Windows PC