Associator of Associative Algebra is Zero

Theorem

Let $\struct {A_R, \oplus}$ be an associative algebra.

Let $\sqbrk {a, b, c}$ denote the associator of $a, b, c \in A_R$.

Then:

$\forall a, b, c \in A_R: \sqbrk {a, b, c} = \mathbf 0_R$

Proof

\(\ds \forall a, b, c \in A_R: \, \)

\(\ds a \oplus \paren {b \oplus c}\)

\(=\)

\(\ds \paren {a \oplus b} \oplus c\)

Definition of Associative Algebra

\(\ds \leadsto \ \ \)

\(\ds a \oplus \paren {b \oplus c} - \paren {a \oplus b} \oplus c\)

\(=\)

\(\ds \mathbf 0_R\)

$\blacksquare$

Sources

• 2002: John C. Baez: The Octonions: 1.1 Preliminaries