The phrase "embodying the idea" is correct and usable in written English.
You can use it when discussing how something represents or gives tangible form to a concept or notion.
Example: "The sculpture is embodying the idea of freedom through its dynamic form and open structure."
Alternatives: "representing the concept" or "capturing the essence."
As an example of her unique spirit, in 1955 as a widow, she adopted a baby, embodying the idea of a modern woman.
And yet the poem comes a little too close to embodying the idea it seems to be dismissing: that poems should coolly express the costs of women's depredations but maintain their own "prancing," elegant distance from violence and terror.
There are so many of us, forever fascinated by the story, who still cannot quite fathom how the band managed to make music so endlessly full of interest, while also embodying the idea that as the world was changing at an unprecedented rate, they were always ahead of everyone else.
Wittgenstein proposal thus mirrors Russell's in embodying the idea that it makes no sense to speak of the existence of immediately given (that is, named) simples (cf. PM, 174 5).
The show's title plays on this a bit, embodying the idea of a facade and exterior, as well as creating a memo for viewers in regard to Piallat's installation structure.
Here was an individual who had been celebrated and touted as a normal, everyday kind of person, and she'd died a tragic death on national TV for audaciously embodying the idea that anyone could go to space.
"Sarge came to embody the idea of public service," President Obama said in a statement.
His designs embody the idea that, in the modern city, taste should be constantly challenged.
Simulation studies on a benchmark and a real-life problem are carried out to embody the idea of the paper.
Christopher Oram's design, with its swinging thurible and high-barred windows, embodies the idea of Spain as a religious prison.
