The Lorenz equation has a natural symmetry defined by

To verify that (1) is a symmetry of an ordinary differential equation (Lorenz equation) there must exist a $3\times 3$ matrix which commutes with the differential equation. This can be easily verified by observing that the symmetry is associated with the matrix $R$ defined as

Let

where $f(\textbf{x})$ is the Lorenz equation and $\textbf{x}^{T}=(x,y,z)$. We proceed by showing that $Rf(\textbf{x})=f(R\textbf{x})$. Looking at the left hand side

and now looking at the right hand side

Since the left hand side is equal to the right hand side then (1) is a symmetry of the Lorenz equation.