i have to show that $\displaystyle \Delta(x) = x \otimes 1 + 1 \otimes x$ is a universal enveloping algebra homomorphism.

do i have to show that it is a universal enveloping algebra, or just straight up that it is a homomorphism?

this is my working to show that it is a homomorphism:

want to show that $\displaystyle \Delta(xy)=\Delta(x)\Delta(y)$

$\displaystyle \Delta(xy)$

$\displaystyle = (xy) \otimes 1 + 1 \otimes (xy)$

$\displaystyle = (x \times 1)(y \otimes 1) + (1 \otimes x)(1 \otimes y)$

this is where i'm stuck because i want to show that it equals

$\displaystyle \Delta(x)\Delta(y)$

$\displaystyle = (x \otimes 1 + 1 \otimes x)(y \otimes 1 + 1 \otimes y)$

$\displaystyle = (x \otimes 1)(y \otimes 1) + (x \otimes 1)(1 \otimes y) + (1 \otimes x)(y \otimes 1) + (1 \otimes x)(1 \otimes y)$

by any chance does $\displaystyle (x \otimes 1)(1 \otimes y) + (1 \otimes x)(y \otimes 1) = 0$? that would mean my working is correct!