 Solution

The first column is five times $(4,0,3,0{)}^{\mathsf{T}}/5$. Its orthogonal companion in the first-and-third-coordinate plane is $(3,0,-4,0{)}^{\mathsf{T}}/5$, and the second column has positive residual seven times that vector. The remaining coordinate directions can therefore be read off immediately rather than performing a general four-dimensional orthogonalization. One obtainsThe columns of $Q$ are visibly orthonormal. Taking their inner products with each column of the given matrix gives the displayed $R$, which also verifies $QR=A$ and positive diagonal entries.