The phrase "a core lesson" is correct and usable in written English.
It can be used when referring to an essential or fundamental teaching or principle that is important to understand.
Example: "One core lesson from this experience is the importance of resilience in the face of challenges."
Alternatives: "a fundamental lesson" or "an essential lesson".
This is a global crisis, and a core lesson of the Great Depression is that a global crisis calls for a global response.
Any group can become Team B. Sakaguchi's experiences underscore a core lesson of Google's research into teamwork: By adopting the data-driven approach of Silicon Valley, Project Aristotle has encouraged emotional conversations and discussions of norms among people who might otherwise be uncomfortable talking about how they feel.
This is the one nobody in the executive suite really wants to hear about, but it's simply a core lesson of business history.
In Tennessee, state education officials offered sessions to about 30,000 teachers this summer, and in Delaware, Mark Murphy, the secretary of education, said that school districts would be hosting "back to school" nights where legislators can "take part in a Common Core lesson to see and experience the type of learning that the students will be getting".
Take note of any core lessons you receive.
That is when I usually have some kind of breakthrough moment; when I learn a core life lesson.
He would also observe the core lesson of this campaign: conservatism loses; moderation wins.
The core lesson of 2016 is that there is nothing inevitable in the march of progress.
One former Israeli military-intelligence officer summarized the core lesson this way: "How to do targeted killing, which is very relevant to the success of the war, and what the United States is going to have to do".
This is the core lesson of the Turkish media outlets' spectacular failure to cover the Gezi Park protests, and their subsequent aggressive outbursts against the international media outlets that chose to cover Turks marching through the streets of Istanbul rather than penguins waddling across the ice of Antarctica.
Ultimately, I think the core lesson is that it's all about people, and so you've got to focus on that to understand what's going on, what the context is and make sure you get people aligned around the right priorities.
