十六元数

在抽象代数中，十六元数（英语：Sedenion）是在实数上形成的16维非交换且非结合代数结构。彷如八元数，其乘法不符合交换律及结合律。十六元数可以透过将八元数套用凯莱-迪克森结构来构造。然而，与八元数不一样，十六元数甚至不符合交错性。尽管如此，十六元数仍然符合幂结合性。此外，十六元数中存在零因子（zero divisor），例如${\displaystyle {{\left({{\,e_{3}}+{\,e_{10}}}\right)}\times {\left({{\,e_{6}}-{\,e_{15}}}\right)}}=0}$，这点与八元数截然不同——因此，十六元数无法构成整环（integral domain），也无法构成除环（divisor ring）。^[1]

十六元数
符号	${\displaystyle \mathbb {S} }$
种类	非结合代数
单位	${\displaystyle e_{0}}$、 ${\displaystyle e_{1}}$、 ${\displaystyle e_{2}}$、 ${\displaystyle e_{3}}$、 ${\displaystyle e_{4}}$、 ${\displaystyle e_{5}}$、 ${\displaystyle e_{6}}$、 ${\displaystyle e_{7}}$、 ${\displaystyle e_{8}}$、 ${\displaystyle e_{9}}$、 ${\displaystyle e_{10}}$、 ${\displaystyle e_{11}}$、 ${\displaystyle e_{12}}$、 ${\displaystyle e_{13}}$、 ${\displaystyle e_{14}}$ 及${\displaystyle e_{15}}$
乘法单位元	${\displaystyle e_{0}}$
主要性质	幂结合性 分配律
数字系统
${\displaystyle \mathbb {N} }$ 自然数 ${\displaystyle \mathbb {Z} }$ 整数 ${\displaystyle \mathbb {Q} }$ 有理数 ${\displaystyle \mathbb {R} }$ 实数 ${\displaystyle \mathbb {C} }$ 复数 ${\displaystyle \mathbb {H} }$ 四元数 ${\displaystyle \mathbb {O} }$ 八元数 ${\displaystyle \mathbb {S} }$ 十六元数 ${\displaystyle \mathbb {T} }$ 三十二元数
查 论 编

各种各样的数 各种各样的数 基本 ${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$ 正数 ${\displaystyle \mathbb {R} ^{+}}$ 自然数 ${\displaystyle \mathbb {N} }$ 正整数 ${\displaystyle \mathbb {Z} ^{+}}$ 小数 有限小数 无限小数 循环小数 有理数 ${\displaystyle \mathbb {Q} }$ 代数数 ${\displaystyle \mathbb {A} }$ 实数 ${\displaystyle \mathbb {R} }$ 复数 ${\displaystyle \mathbb {C} }$ 高斯整数 ${\displaystyle \mathbb {Z} [i]}$ 负数 ${\displaystyle \mathbb {R} ^{-}}$ 整数 ${\displaystyle \mathbb {Z} }$ 负整数 ${\displaystyle \mathbb {Z} ^{-}}$ 分数 单位分数 二进分数 规矩数 无理数 超越数 虚数 ${\displaystyle \mathbb {I} }$ 二次无理数 艾森斯坦整数 ${\displaystyle \mathbb {Z} [\omega ]}$ 延伸 二元数 四元数 ${\displaystyle \mathbb {H} }$ 八元数 ${\displaystyle \mathbb {O} }$ 十六元数 ${\displaystyle \mathbb {S} }$ 超实数 ${\displaystyle ^{*}\mathbb {R} }$ 大实数 上超实数 双曲复数 双复数 复四元数 共四元数（英语：Dual quaternion） 超复数 超数 超现实数 其他 素数 ${\displaystyle \mathbb {P} }$ 可计算数 基数 阿列夫数 同余 整数数列 公称值 规矩数 可定义数 序数 超限数 p进数 数学常数 圆周率 ${\displaystyle \pi =3.14159265}$… 自然对数的底 ${\displaystyle e=2.718281828}$… 虚数单位 ${\displaystyle i={\sqrt {-{1}}}}$ 无限大 ${\displaystyle \infty }$ 查 论 编

十六元数是由八元数套用凯莱-迪克森构造而成的。十六元数亦可以继续进行凯莱-迪克森构造。若将十六元数套用凯莱-迪克森构造将会形成三十二元数（trigintaduonion）。^[2]每一次的构造都会导致维数翻倍^[3]:45，并且构造结果同样与十六元数类似，有着不符合交错性、符合幂结合性与存在零因子等特性。^[3]

十六元数这个术语同时亦用于其他同为16维度的代数结构，例如两个复四元数的张量积、实数上的4×4矩阵代数或乔纳森·D·H·史密斯于1995提出的一种代数结构。^[4]

算术

立方八元数扩展到四维空间的视觉化^[5]:6，其展示了35个示例十六元数之实数${\displaystyle (e_{0})}$顶点三元组所构成的超平面。唯一的例外是三元组${\displaystyle (e_{1})}$, ${\displaystyle (e_{2})}$, ${\displaystyle (e_{3})}$不与${\displaystyle (e_{0})}$形成超平面。

十六元数的乘法和八元数一样，不具备交换律及结合律。与八元数不同的是，十六元数不具备交错代数的特性。虽然如此，但十六元数仍然保有幂结合性，也就是说，对所有的十六元数集${\displaystyle \mathbb {S} }$中的元素x，幂${\displaystyle x^{n}}$是可以明确定义的。同时，十六元数亦有柔性代数（英语：Flexible algebra）的特性。^[6]

十六元数共有的16个单位。这16个单位十六元数是：^[7]

${\displaystyle e_{0}}$、${\displaystyle e_{1}}$、${\displaystyle e_{2}}$、${\displaystyle e_{3}}$、${\displaystyle e_{4}}$、${\displaystyle e_{5}}$、${\displaystyle e_{6}}$、${\displaystyle e_{7}}$、${\displaystyle e_{8}}$、${\displaystyle e_{9}}$、${\displaystyle e_{10}}$、${\displaystyle e_{11}}$、${\displaystyle e_{12}}$、${\displaystyle e_{13}}$、${\displaystyle e_{14}}$及${\displaystyle e_{15}}$

每个十六元数都是单位十六元数${\displaystyle e_{0}}$, ${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$, ${\displaystyle e_{3}}$, ..., ${\displaystyle e_{15}}$的线性组合，并构成了十六元数向量空间的基。 每个十六元数都可以用以下形式表示：^[7]

${\displaystyle x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\cdots +x_{14}e_{14}+x_{15}e_{15}.}$

十六元数的加法和减法是通过将相应十六元数单位之系数的加法或减法来定义的。而十六元数的乘法是对加法的分配，所以两个十六元数的乘积可以通过对所有项的乘积求和来计算。^[1]

十六元数和其他也由凯莱-迪克森结构来构造的代数结构一样，其皆包含了依凯莱-迪克森结构构造来源的代数结构。例如十六元数可透过八元数代凯莱-迪克森结构来构造、八元数可透过四元数代凯莱-迪克森结构来构造、四元数可透过复数代凯莱-迪克森结构来构造、复数可透过实数代凯莱-迪克森结构来构造。因此，十六元数系包含了一个八元数系（由下方乘法表对应的${\displaystyle e_{0}}$至${\displaystyle e_{7}}$构造），亦包含了四元数系（由${\displaystyle e_{0}}$至${\displaystyle e_{3}}$构造），也包含了复数系（由${\displaystyle e_{0}}$至${\displaystyle e_{1}}$构造）和实数系（由${\displaystyle e_{0}}$构造）。^[8]

十六元数具有乘法单位元${\displaystyle e_{0}}$和乘法逆元，但因为存在零因子因此无法构成可除代数（英语：Division algebra）。换句话说，即十六元数的代数系统中，存在2个非零十六元数相乘为零，例如${\displaystyle (e_{3}+e_{10})(e_{6}-e_{15})=0}$。其他基于凯莱-迪克森结构构造的超复数系统中，维度大于16的超复数也都存在零因子。^[7][1]

十六元数单元乘法表如下：^[8]

${\displaystyle e_{i}e_{j}}$		${\displaystyle e_{j}}$
		${\displaystyle e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle e_{8}}$	${\displaystyle e_{9}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$
${\displaystyle e_{i}}$	${\displaystyle e_{0}}$	${\displaystyle e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle e_{8}}$	${\displaystyle e_{9}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$
	${\displaystyle e_{1}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$
	${\displaystyle e_{2}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{4}}$	${\displaystyle -e_{5}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$
	${\displaystyle e_{3}}$	${\displaystyle e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$	${\displaystyle e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$
	${\displaystyle e_{4}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{6}}$	${\displaystyle -e_{7}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{11}}$
	${\displaystyle e_{5}}$	${\displaystyle e_{5}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{8}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$
	${\displaystyle e_{6}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{1}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle -e_{8}}$	${\displaystyle e_{9}}$
	${\displaystyle e_{7}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$	${\displaystyle e_{5}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{8}}$
	${\displaystyle e_{8}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle -e_{12}}$	${\displaystyle -e_{13}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$
	${\displaystyle e_{9}}$	${\displaystyle e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{5}}$	${\displaystyle e_{4}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$
	${\displaystyle e_{10}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{6}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$
	${\displaystyle e_{11}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$	${\displaystyle e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle -e_{5}}$	${\displaystyle e_{4}}$
	${\displaystyle e_{12}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{2}}$	${\displaystyle -e_{3}}$
	${\displaystyle e_{13}}$	${\displaystyle e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$	${\displaystyle e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{2}}$
	${\displaystyle e_{14}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle e_{8}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{6}}$	${\displaystyle -e_{7}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$
	${\displaystyle e_{15}}$	${\displaystyle e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$

十六元数特性

从上表可得到：

对所有的${\displaystyle i}$，有${\displaystyle e_{0}e_{i}=e_{i}e_{0}=e_{i}}$，

${\displaystyle e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0}$，且

${\displaystyle e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{with}}\,\,i,j\neq 0}$。

反结合

十六元数并非完全反结合。选择任意四个生成元${\displaystyle i,j,k}$和${\displaystyle l}$，对于乘积${\displaystyle ijkl}$，有五种添加括号的方法。假如反结合律总是成立，则五者之间应有以下关系：

${\displaystyle (ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl),}$

从而${\displaystyle (ij)(kl)=0}$，矛盾。所以，某两者之间不满足反结合律。

特别地，代入${\displaystyle e_{1},e_{2},e_{4}}$和${\displaystyle e_{8}}$时，利用上列乘法表，可得最后两式满足结合律：${\displaystyle e_{1}(e_{2}e_{12})=(e_{1}e_{2})e_{12}=-e_{15}}$。

四元子代数

在下表列出了构成这个特定十六元数乘法表的35个三元组。用于使用凯莱-迪克森结构构造之十六元数的7个八元数三元组，以粗体表示：

每个三元组中，三个数的二进制表示，按位异或的结果为0。

{ {1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15},
{2, 4, 6}, {2, 5, 7}, {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, {3, 4, 7},
{3, 6, 5}, {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},
{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},
{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10} }

84组由十六元数单位组成的零因子数组${\displaystyle \{e_{a},e_{b},e_{c},e_{d}\}}$列举如下，其中 ${\displaystyle (e_{a}+e_{b})\circ (e_{c}+e_{d})=0}$：

${\displaystyle {\begin{array}{c}{\begin{array}{ccc}1\leq a\leq 6,&c>a,&9\leq b\leq 15\\9\leq d\leq 15&&-9\geq d\geq -15\end{array}}\\{\begin{array}{ll}\{e_{1},e_{10},e_{5},e_{14}\}&\{e_{1},e_{10},e_{4},-e_{15}\}\\\{e_{1},e_{10},e_{7},e_{12}\}&\{e_{1},e_{10},e_{6},-e_{13}\}\\\{e_{1},e_{11},e_{4},e_{14}\}&\{e_{1},e_{11},e_{6},-e_{12}\}\\\{e_{1},e_{11},e_{5},e_{15}\}&\{e_{1},e_{11},e_{7},-e_{13}\}\\\{e_{1},e_{12},e_{2},e_{15}\}&\{e_{1},e_{12},e_{3},-e_{14}\}\\\{e_{1},e_{12},e_{6},e_{11}\}&\{e_{1},e_{12},e_{7},-e_{10}\}\\\{e_{1},e_{13},e_{6},e_{10}\}&\{e_{1},e_{13},e_{7},-e_{14}\}\\\{e_{1},e_{13},e_{7},e_{11}\}&\{e_{1},e_{13},e_{3},-e_{15}\}\\\{e_{1},e_{14},e_{2},e_{13}\}&\{e_{1},e_{14},e_{4},-e_{11}\}\\\{e_{1},e_{14},e_{3},e_{12}\}&\{e_{1},e_{14},e_{5},-e_{10}\}\\\{e_{1},e_{15},e_{3},e_{13}\}&\{e_{1},e_{15},e_{2},-e_{12}\}\\\{e_{1},e_{15},e_{4},e_{10}\}&\{e_{1},e_{15},e_{5},-e_{11}\}\\\{e_{2},e_{9},e_{4},e_{15}\}&\{e_{2},e_{9},e_{5},-e_{14}\}\\\{e_{2},e_{9},e_{6},e_{13}\}&\{e_{2},e_{9},e_{7},-e_{12}\}\\\{e_{2},e_{11},e_{5},e_{12}\}&\{e_{2},e_{11},e_{4},-e_{13}\}\\\{e_{2},e_{11},e_{6},e_{15}\}&\{e_{2},e_{11},e_{7},-e_{14}\}\\\{e_{2},e_{12},e_{3},e_{13}\}&\{e_{2},e_{12},e_{5},-e_{11}\}\\\{e_{2},e_{12},e_{7},e_{9}\}&\{e_{2},e_{13},e_{3},-e_{12}\}\\\{e_{2},e_{13},e_{4},e_{11}\}&\{e_{2},e_{13},e_{6},-e_{9}\}\\\{e_{2},e_{14},e_{5},e_{9}\}&\{e_{2},e_{14},e_{3},-e_{15}\}\\\{e_{2},e_{14},e_{3},e_{14}\}&\{e_{2},e_{15},e_{4},-e_{9}\}\\\{e_{2},e_{15},e_{3},e_{14}\}&\{e_{2},e_{15},e_{6},-e_{11}\}\\\{e_{3},e_{9},e_{6},e_{12}\}&\{e_{3},e_{9},e_{4},-e_{14}\}\\\{e_{3},e_{9},e_{7},e_{13}\}&\{e_{3},e_{9},e_{5},-e_{15}\}\\\{e_{3},e_{10},e_{4},e_{13}\}&\{e_{3},e_{10},e_{5},-e_{12}\}\\\{e_{3},e_{10},e_{7},e_{14}\}&\{e_{3},e_{10},e_{6},-e_{15}\}\\\{e_{3},e_{12},e_{5},e_{10}\}&\{e_{3},e_{12},e_{6},-e_{9}\}\\\{e_{3},e_{14},e_{4},e_{9}\}&\{e_{3},e_{13},e_{4},-e_{10}\}\\\{e_{3},e_{15},e_{5},e_{9}\}&\{e_{3},e_{13},e_{7},-e_{9}\}\\\{e_{3},e_{15},e_{6},e_{10}\}&\{e_{3},e_{14},e_{7},-e_{10}\}\\\{e_{4},e_{9},e_{7},e_{10}\}&\{e_{4},e_{9},e_{6},-e_{11}\}\\\{e_{4},e_{10},e_{5},e_{11}\}&\{e_{4},e_{10},e_{7},-e_{9}\}\\\{e_{4},e_{11},e_{6},e_{9}\}&\{e_{4},e_{11},e_{5},-e_{10}\}\\\{e_{4},e_{13},e_{6},e_{15}\}&\{e_{4},e_{13},e_{7},-e_{14}\}\\\{e_{4},e_{14},e_{7},e_{13}\}&\{e_{4},e_{14},e_{5},-e_{15}\}\\\{e_{4},e_{15},e_{5},e_{14}\}&\{e_{4},e_{15},e_{6},-e_{13}\}\\\{e_{5},e_{10},e_{6},e_{9}\}&\{e_{5},e_{9},e_{6},-e_{10}\}\\\{e_{5},e_{11},e_{7},e_{9}\}&\{e_{5},e_{9},e_{7},-e_{11}\}\\\{e_{5},e_{12},e_{7},e_{14}\}&\{e_{5},e_{12},e_{6},-e_{15}\}\\\{e_{5},e_{15},e_{6},e_{12}\}&\{e_{5},e_{14},e_{7},-e_{12}\}\\\{e_{6},e_{11},e_{7},e_{10}\}&\{e_{6},e_{10},e_{7},-e_{11}\}\\\{e_{6},e_{13},e_{7},e_{12}\}&\{e_{6},e_{10},e_{7},-e_{13}\}\end{array}}\end{array}}}$

应用

莫雷诺·吉列尔莫于1998年表明，一对范数一的十六元数空间（每个元素皆为范数为1的十六元数二元组的空间）中的元素相乘为零这样的代数空间与紧凑形式的例外李群G₂（英语：G2 (mathematics)）同胚。^[7]（留意在莫雷诺论文中，零因子指的是一对相乘为零的元素。）

十六元数神经网络在机器学习应用中提供了一种高效且紧凑的表达方式，并被用于解决多个时间序列预测问题。^[9]

参见

• 超复数
• 伦纳德·尤金·迪克森