Editing Sedenion (section)

=== Multiplication ===
Like [[octonion]]s, [[multiplication]] of sedenions is neither [[commutative]] nor [[associative]]. However, in contrast to the octonions, the sedenions do not even have the property of being [[alternative algebra|alternative]]. They do, however, have the property of [[power associativity]], which can be stated as that, for any element <math>x</math> of <math>\mathbb{S}</math>, the power <math>x^n</math> is well defined. They are also [[Flexible algebra|flexible]].

The sedenions have a multiplicative [[identity element]] <math>e_0</math> and multiplicative inverses, but they are not a [[division algebra]] because they have [[zero divisors]]: two nonzero sedenions can be multiplied to obtain zero, for example <math>(e_3 + e_{10})(e_6 - e_{15})</math>. All [[hypercomplex number]] systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

The sedenion multiplication table is shown below:
{| class="wikitable" style="margin:1em auto; text-align: center;"
  !colspan="2" rowspan="2"| <math>e_ie_j</math>
  !colspan="16" |<math>e_j</math>
|-
  ! <math>e_0</math>
  ! <math>e_1</math>
  ! <math>e_2</math>
  ! <math>e_3</math>
  ! <math>e_4</math>
  ! <math>e_5</math>
  ! <math>e_6</math>
  ! <math>e_7</math>
  ! <math>e_8</math>
  ! <math>e_9</math>
  ! <math>e_{10}</math>
  ! <math>e_{11}</math>
  ! <math>e_{12}</math>
  ! <math>e_{13}</math>
  ! <math>e_{14}</math>
  ! <math>e_{15}</math>
|- <!-- Would color coding the cells be nice or annoying? There are too many similar greenish shades though... -->
  ! rowspan="16" | <math>e_i</math>
  ! width="30pt" | <math>e_0</math><!-- color: white; when minus sign -->
  | width="30pt" <!-- style="color: black; background-color: hsla(  0,   0%, 50%, 1);" --> | <math>e_0</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(  0, 100%, 50%, 1);" --> | <math>e_1</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 16, 100%, 50%, 1);" --> | <math>e_2</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 32, 100%, 50%, 1);" --> | <math>e_3</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 48, 100%, 50%, 1);" --> | <math>e_4</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 64, 100%, 50%, 1);" --> | <math>e_5</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 80, 100%, 50%, 1);" --> | <math>e_6</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 96, 100%, 50%, 1);" --> | <math>e_7</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(112, 100%, 50%, 1);" --> | <math>e_8</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(128, 100%, 50%, 1);" --> | <math>e_9</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(144, 100%, 50%, 1);" --> | <math>e_{10}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(160, 100%, 50%, 1);" --> | <math>e_{11}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(176, 100%, 50%, 1);" --> | <math>e_{12}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(192, 100%, 50%, 1);" --> | <math>e_{13}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(208, 100%, 50%, 1);" --> | <math>e_{14}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(224, 100%, 50%, 1);" --> | <math>e_{15}</math>
|-
  ! <math>e_1</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_3</math>
  | <math>-e_2</math>
  | <math>e_5</math>
  | <math>-e_4</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>-e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
|-
  ! <math>e_2</math>
  | <math>e_2</math>
  | <math>-e_3</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>-e_4</math>
  | <math>-e_5</math>
  | <math>e_{10}</math>
  | <math>e_{11}</math>
  | <math>-e_8</math>
  | <math>-e_9</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
|-
  ! <math>e_3</math>
  | <math>e_3</math>
  | <math>e_2</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_5</math>
  | <math>-e_4</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>-e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
|-
  ! <math>e_4</math>
  | <math>e_4</math>
  | <math>-e_5</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_2</math>
  | <math>e_3</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{14}</math>
  | <math>e_{15}</math>
  | <math>-e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
|-
  ! <math>e_5</math>
  | <math>e_5</math>
  | <math>e_4</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>-e_3</math>
  | <math>e_2</math>
  | <math>e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
|-
  ! <math>e_6</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>e_4</math>
  | <math>-e_5</math>
  | <math>-e_2</math>
  | <math>e_3</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_8</math>
  | <math>e_9</math>
|-
  ! <math>e_7</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_5</math>
  | <math>e_4</math>
  | <math>-e_3</math>
  | <math>-e_2</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_9</math>
  | <math>-e_8</math>
|-
  ! <math>e_8</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_{12}</math>
  | <math>-e_{13}</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_2</math>
  | <math>e_3</math>
  | <math>e_4</math>
  | <math>e_5</math>
  | <math>e_6</math>
  | <math>e_7</math>
|-
  ! <math>e_9</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>-e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>-e_3</math>
  | <math>e_2</math>
  | <math>-e_5</math>
  | <math>e_4</math>
  | <math>e_7</math>
  | <math>-e_6</math>
|-
  ! <math>e_{10}</math>
  | <math>e_{10}</math>
  | <math>e_{11}</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>-e_2</math>
  | <math>e_3</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>e_4</math>
  | <math>e_5</math>
|-
  ! <math>e_{11}</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>-e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>-e_3</math>
  | <math>-e_2</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_5</math>
  | <math>e_4</math>
|-
  ! <math>e_{12}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{14}</math>
  | <math>e_{15}</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_4</math>
  | <math>e_5</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>-e_2</math>
  | <math>-e_3</math>
|-
  ! <math>e_{13}</math>
  | <math>e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>-e_5</math>
  | <math>-e_4</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_3</math>
  | <math>-e_2</math>
|-
  ! <math>e_{14}</math>
  | <math>e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{10}</math>
  | <math>-e_{11}</math>
  | <math>e_8</math>
  | <math>e_9</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>-e_4</math>
  | <math>e_5</math>
  | <math>e_2</math>
  | <math>-e_3</math>
  | <math>-e_0</math>
  | <math>e_1</math>
|-
  ! <math>e_{15}</math>
  | <math>e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_9</math>
  | <math>e_8</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_5</math>
  | <math>-e_4</math>
  | <math>e_3</math>
  | <math>e_2</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
|}