Caching intermediary CTEs for multiple uses

I'm working on this huge query that I chose to use CTEs. Due to the complexity of this query, I ended up having to reuse intermediary query results more than once in subsequent queries. The problem I'm facing can be summarized by the following toy code:

WITH cte1 AS (    SELECT id, name    FROM Manager), cte2 AS (    SELECT id, name    FROM Employee     LEFT OUTER JOIN cte1 ON Employee.id = cte1.id)SELECT *FROM cte1

I have notice an unusual amount of indexes seeks in a fairly small table that was being queried in one the first queries which, according to the execution plan, were taking about 30% of the whole query. This led to me believe the base table was being queried multiple times as it was being referenced more than once in subsequent queries.

I was under the impression that intermediary results in a CTE were cached until the final statement. Reading Microsofts docs on CTEs, I couldn't find any reason to think otherwise.

Are intermediary CTEs cached for muliple uses within an WITH statement? If not, is there a way to cache it without rewriting the query?

Thanks!