Definition 30.2.2 (Normal).label Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then the following are equivalent:

1. (1)

   $\text{Re}(x)\text{Im}(x) = \text{Im}(x)\text{Re}(x)$.

2. (2)

   $x^{*}x = xx^{*}$.

If the above holds, then $x$ is normal.