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We show that if the general real method 
${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$.
We present “reiteration theorems” with limiting values 
$\theta =0$ and 
$\theta =1$ for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in 
$[\text{D}]$.
