> Similarly for the second component.

Hmm.

{a + b, c + d} = {a, c + d} + {b, c + d} = {a, c} + {a, d} + {b, c} + {b, d}.

{a + b + c, x + y + z} = {a, x + y + z} + {b, x + y + z} + {c, x + y + z} = (a sum of nine direct brackets).

This doesn't look like it will scale well.

Then don't use the Lie bracket. All bilinear forms scale the same way.