By a cubic curve in \(SE\left ( 3\right ) \) will be understood a frame motion expressed with the exponential map in terms of canonical (exponential) coordinates that are cubic in the path parameter. A cubic POE spline on \( SE\left ( 3\right ) \) is a product of exponentials, each defined on a segment of the curve, where within each segment the curve is described by canonical coordinates that are cubic in the path parameter, and the spline satisfies second-order continuity conditions. Such POE splines were proposed several years ago. It is most obvious to assume that such a spline is able to at least exactly reconstruct a given cubic curve. In this paper, it is shown that this is not the case, however. It is further shown that this assumption holds true when the interpolated curve defines a one-parameter subgroup. An alternative formulation of the POE spline is proposed that does not suffer from this shortcoming.