<p><it>Abstract—</it>In this paper, we introduce a new method to reduce a real matrix to a real Schur form by a sequence of similarity transformations that are 3-D orthogonal transformations. Two significant features of this method are that (1) all the transformed matrices and all the computations are done in the real field, and (2) it can be easily parallelized. We call the algorithm that uses this method the real two-zero (RTZ) algorithm. We describe in this paper both serial and parallel implementations of the RTZ algorithm. Our tests indicate that the rate of convergence to a real Schur form is quadratic for real near-normal matrices with real distinct Eigenvalues. Suppose $<tmath>n</tmath>$ is the order of a real matrix $<tmath>A</tmath>$. In order to choose a sequence of 3-D orthogonal transformations on $<tmath>A</tmath>$, we need to determine some ordering on triples in $<tmath>{\cal T} = \{(k,l,m)\mid 1 \leq k \char'74 l \char'74 m \leq n\}</tmath>$, where $<tmath>(k,l,m)</tmath>$ defines the three coordinates under the 3-D transformation. We show how the ordering of the triples used in our implementations can be generated cyclically in an algorithm.</p><p><it>Index Terms—</it>Real two-zero algorithm, 3-D orthogonal transformations, ordering on triples, real Schur form, algorithm to generate triples, householder matrix, modified Gao–Thomas algorithm, quadratic convergence, real field</p>