After login gitlab, is there any sso token in cookie can use for other application SSO login?

I have set up our own gitlab server and use it as oauth IDP. the steps is :

1. open my Application and click GitLab as an IDP for an OAuth login.
2. my Application redirects you to GitLab
3. login gitlab. it redirects you to your App and the redirect contains a token which your app then can use to check validity of the login.

now I want to do a new feature:

1. the user login gitlab first
2. the user open my application in the same browser.
3. my application can use the gitlab’s cookie or session or token to login

I want to find some cookie which may contain some sso token. so my application can retrieve it from cookie and try to validate via gitlab. but seems that there is not cookie except _gitlab_session

so my questions is :

1. is there any sso token I can use for other application SSO login after login in gitlab ?
2. is there a api to validate the cookie/token ?