\ begin {alinear*} [a, b] [e, f] + [c, d] [e, f] & = [a e, b f] + [c e, d f]\\ & = [a e d f + b f c e, b d f^2]\\ & = [a e d + b c e, b d f]\\ & = [a d e + b c e, b d f]\\ & = ([a, b] + [c, d]) [e, f]...\ begin {alinear*} [a, b] [e, f] + [c, d] [e, f] & = [a e, b f] + [c e, d f]\\ & = [a e d f + b f c e, b d f^2]\\ & = [a e d + b c e, b d f]\\ & = [a d e + b c e, b d f]\\ & = ([a, b] + [c, d]) [e, f]\ final {alinear*}
