In the [abstract index notation of Penrose](https://en.wikipedia.org/wiki/Abstract_index_notation), indicies serve as placeholders to indicate the type of a tensor field. For example, $X^i$ denotes a vector field. What is the commonly accepted notation for the Lie bracket of two vector fields $X^i$ and $Y^j$?
Clearly, $[X^i, Y^j]^k$ does not work, because this would denote a $3$-contravariant tensor. Something like $[ \cdot, \cdot]\_{ij}^{\ \ k} X^i Y^j$ would work but looks strange. 



Lie bracket of vector fields in Penrose's abstract index notation
Penrose-Rindler write it $X^i\nabla\_iY^j-Y^i\nabla\_iX^j$.


