"laid a cornerstone" is correct and usable in written English.
You can use it to refer to an event during which a stone is ceremonially set to mark the beginning of a building project. For example: "Yesterday, the mayor laid a cornerstone for the new library, marking the start of its construction."
Right-wing officials and activists, accompanied by Mike Huckabee, the Republican former governor of Arkansas, laid a cornerstone on Tuesday for new Jewish housing on the Mount of Olives in contested East Jerusalem.
Working at a company was not my first choice, but I gave it everything I had, and the work I did there laid a cornerstone for my future success.
As it is, he has laid a cornerstone for a book that remains to be written, and that may in time occupy the same shelf as Machiavelli, Hobbes, Rousseau, and Dewey.
At noon, Mr. Hikind led a group of about 50 American Jews in laying a cornerstone for the next phase of Nof Zion, with construction scheduled to start next spring.
It related bills that he has introduced and stated that he was a "proponent for the continuation of building throughout Jerusalem," mentioning that he had participated in laying a cornerstone of a new Jewish neighborhood in East Jerusalem.
The king laid a crystal cornerstone into a stainless steel shaft on wheels.
Understanding the susceptibility or resistance to diseases and infections thus laid an early cornerstone for the creation of predictive medicine through personalized disease prevention.
Over the years he laid a number of cornerstones that shaped the field.
Now, he said, "we're laying a different cornerstone".
The rebels laid an ambush, the cornerstone of their guerrilla tactics and their only way to get weapons and supplies (the trucks that ferried us around for three days had been captured from the government a few weeks earlier).
Before he retired as chief justice in 2005, he laid the cornerstone for a new home for the Constitutional Court, the site of a dilapidated prison that once held South African freedom fighters.
