I am new to the study of Lie algebras and particularly algebraic Lie algebras. In the paper
Chevalley, Claude, Algebraic Lie algebras, Ann. Math. (2) 48, 91-100 (1947). ZBL0032.25202.
the author states
Let ${h}$ be an abelian subalgebra of $gl(n;K)$ which has a basis ${B_1,B_2,,,,,B_r}$ composed of semisimple matrices. Then every matrix in ${h}$ is semisimple and the identity matrix of ${h}$ into $gl(n;K)$ is a semisimple representation of ${h}$.
The proof begins with Let ${B}$ be the associative algebra generated by the elements of ${h}$ and unit matrix E. I do not understand how to generate an associative algebra when the multiplication is already predefined in the algebra. Any explanation would be of great help.