Understanding Frobenius' Theorem on real division algebras

Question

The proof I am reading is this one here. I don't understand the following:

"Any $d$ in $D$ defines an endomorphism of $d$ by left multiplication".

What endomorphism is defined, and how?

Answer 1

The endomorphism is $D\rightarrow D$, $x\mapsto dx$. The word "endomorphism" here means "endomorphism of a real vector space" not "endomorphism of a ring": left multiplication obviously is not a ring homomorphism.