The sentence comes from pag. 18 of Garling's book "Clifford Algebras: An introduction".
If $A$ is unital, then a subalgebra $B$ is a unital subalgebra if the identity element of $A$ belongs to $B$. For example, if $A$ is a unital algebra then the set $End(A)$ of unital endomorphisms of $A$ is a unital subalgebra of $L(A)$.
Here $A$ is a $K$-algebra, with $\mathbb{R}$ or $\mathbb{C}$ as $K$. $L(A)$ are the set of endomorphisms of $A$ as a vector field, that is seen to be itself e vector field of dimension $dim(A)^2$. $End(A)$ are the set of endomorphisms of $A$ as a $K$-algebra that map $1_A$ to itself.
The sentence claims $End(A)$ to be a linear subspace of $L(A)$, but it is not possible, because if $S$ and $T$ are in $End(A)$ then $(S+T)(1_A)=S(1_A)+T(1_A)=1_A+1_A=2_A$, therefore $S+T$ can't be in $End(A)$.
What is the proper way to define an algebra over the unital endomorphisms of an algebra? Maybe you are able to spot some common flaws in the way I'm approaching this, thank you.