Like what Michael said, there are a number of companies willing to act as an adapter for your SSO needs. My employer, FusionAuth, is another.

Agreed that every SP is slightly different. There's a reason the SAML spec is 800+ pages, it covers a lot of variations. Since it has not been substantially updated since, there are some legacy choices (hello XML).

We also maintain an open source java SAML library: https://github.com/FusionAuth/fusionauth-samlv2

SSO is simple on the surface and complex once you peel it back!