The product of two differential graded Lie algebras, L\times L^\prime , is defined as follows: take the direct sum of the two graded vector spaces L\oplus L^\prime , now equip it with the bracket [(x,x^\prime),(y,y^\prime)]=([x,y],[x^\prime,y^\prime]) and differential D(x,x^\prime )=(dx,d^\prime x^\prime) .
The coproduct of two differential graded Lie algebras, L*L^\prime, is often called the free product.