We describe the use of a guided, transformational approach to systolic array synthesis, using algebraic properties of the operators, such as associativity and commutativity of operands. While there are well known, constructive methods for systolic array synthesis, they do not typically permit the kinds of transformations that we illustrate. We thus expand the design space, to the point of deriving implementations where the standard techniques fail to find any. Obviously, our method needs user guidance, but we develop a characterization that enables us to avoid arbitrary program restructuring.