Lie derivatives. Involutivity

Note that Lgh(x) is itself a scalar function of x and that Lgk(x) IR — IR. Consequently, we can calculate its directional derivative along the vector function g, as 

Similarly, higher-order Lie derivatives can be defined recursively as 

Since the Lie bracket is also a vector function, iterated Lie brackets can be defined using the following standard notation  

The notion of involutivity has also been employed in the statements of Theorem A.l, and throughout the text. It is defined below. 

---