Unitization of Commutative Algebra over Field is Commutative

Theorem
Let $K$ be a field.

Let $A$ be a commutative algebra over $K$.

Let $A_+$ be the unitization of $A$.

Then $A_+$ is commutative.

Proof
Let $\tuple {x, \lambda}, \tuple {y, \mu} \in A_+$.

Then, we have:

So $A_+$ is commutative.