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In , the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the , usually represented by the capital letter S, boldface or \mathbb S.

The sedenions are obtained by applying the Cayley–Dickson construction to the , which can be mathematically expressed as \mathbb{S}=\mathcal{CD}(\mathbb{O},1). As such, the octonions are to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, called the or sometimes the 32-nions.Raoul E. Cawagas, et al. (2009). "THE BASIC SUBALGEBRA STRUCTURE OF THE CAYLEY-DICKSON ALGEBRA OF DIMENSION 32 (TRIGINTADUONIONS)".

The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the , or the algebra of 4 × 4 matrices over the real numbers, or that studied by .

Arithmetic
Every sedenion is a linear combination of the unit sedenions e_0, e_1, e_2, e_3, ..., e_{15}, which form a basis of the of sedenions. Every sedenion can be represented in the form

x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_{14} e_{14} + x_{15} e_{15}.

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.

Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra they were constructed from. So they contain the octonions (generated by e_0 to e_7 in the table below), and therefore also the (generated by e_0 to e_3), (generated by e_0 and e_1) and real numbers (generated by e_0).

Multiplication
Like , of sedenions is neither nor . However, in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of power associativity, which can be stated as that, for any element x of \mathbb{S}, the power x^n is well defined. They are also .

The sedenions have a multiplicative e_0 and multiplicative inverses, but they are not a because they have : two nonzero sedenions can be multiplied to obtain zero, for example (e_3 + e_{10})(e_6 - e_{15}). 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:

!colspan="2" rowspan="2">e_ie_j !colspan="16"e_j 
 ! e_0
 ! e_1
 ! e_2
 ! e_3
 ! e_4
 ! e_5
 ! e_6
 ! e_7
 ! e_8
 ! e_9
 ! e_{10}
 ! e_{11}
 ! e_{12}
 ! e_{13}
 ! e_{14}
 ! e_{15}
! rowspan="16" >e_i ! width="30pt"e_0e_0e_1e_2e_3e_4e_5e_6e_7e_8e_9e_{10}e_{11}e_{12}e_{13}e_{14}e_{15} 
 ! e_1
e_1-e_0e_3-e_2e_5-e_4-e_7e_6e_9-e_8-e_{11}e_{10}-e_{13}e_{12}e_{15}-e_{14} 
 ! e_2
e_2-e_3-e_0e_1e_6e_7-e_4-e_5e_{10}e_{11}-e_8-e_9-e_{14}-e_{15}e_{12}e_{13} 
 ! e_3
e_3e_2-e_1-e_0e_7-e_6e_5-e_4e_{11}-e_{10}e_9-e_8-e_{15}e_{14}-e_{13}e_{12} 
 ! e_4
e_4-e_5-e_6-e_7-e_0e_1e_2e_3e_{12}e_{13}e_{14}e_{15}-e_8-e_9-e_{10}-e_{11} 
 ! e_5
e_5e_4-e_7e_6-e_1-e_0-e_3e_2e_{13}-e_{12}e_{15}-e_{14}e_9-e_8e_{11}-e_{10} 
 ! e_6
e_6e_7e_4-e_5-e_2e_3-e_0-e_1e_{14}-e_{15}-e_{12}e_{13}e_{10}-e_{11}-e_8e_9 
 ! e_7
e_7-e_6e_5e_4-e_3-e_2e_1-e_0e_{15}e_{14}-e_{13}-e_{12}e_{11}e_{10}-e_9-e_8 
 ! e_8
e_8-e_9-e_{10}-e_{11}-e_{12}-e_{13}-e_{14}-e_{15}-e_0e_1e_2e_3e_4e_5e_6e_7 
 ! e_9
e_9e_8-e_{11}e_{10}-e_{13}e_{12}e_{15}-e_{14}-e_1-e_0-e_3e_2-e_5e_4e_7-e_6 
 ! e_{10}
e_{10}e_{11}e_8-e_9-e_{14}-e_{15}e_{12}e_{13}-e_2e_3-e_0-e_1-e_6-e_7e_4e_5 
 ! e_{11}
e_{11}-e_{10}e_9e_8-e_{15}e_{14}-e_{13}e_{12}-e_3-e_2e_1-e_0-e_7e_6-e_5e_4 
 ! e_{12}
e_{12}e_{13}e_{14}e_{15}e_8-e_9-e_{10}-e_{11}-e_4e_5e_6e_7-e_0-e_1-e_2-e_3 
 ! e_{13}
e_{13}-e_{12}e_{15}-e_{14}e_9e_8e_{11}-e_{10}-e_5-e_4e_7-e_6e_1-e_0e_3-e_2 
 ! e_{14}
e_{14}-e_{15}-e_{12}e_{13}e_{10}-e_{11}e_8e_9-e_6-e_7-e_4e_5e_2-e_3-e_0e_1 
 ! e_{15}
e_{15}e_{14}-e_{13}-e_{12}e_{11}e_{10}-e_9e_8-e_7e_6-e_5-e_4e_3e_2-e_1-e_0

Sedenion properties
From the above table, we can see that:

e_0e_i = e_ie_0 = e_i \, \text{for all} \, i,

e_ie_i = -e_0 \,\, \text{for}\,\, i \neq 0, and

e_ie_j = -e_je_i \,\, \text{for}\,\, i \neq j \,\,\text{with}\,\, i,j \neq 0.

Anti-associative
The sedenions are not fully anti-associative. Choose any four generators, i,j,k and l. The following 5-cycle shows that these five relations cannot all be anti-associative.

(ij)(kl) = -((ij)k)l = (i(jk))l = -i((jk)l) = i(j(kl)) = -(ij)(kl)

In particular, in the table above, using e_1,e_2,e_4 and e_8 the last expression associates. (e_1e_2)e_{12} = e_1(e_2e_{12}) = -e_{15}

Quaternionic subalgebras
The particular sedenion multiplication table shown above is represented by 35 triads. The table and its triads have been constructed from an represented by the bolded set of 7 triads using Cayley–Dickson construction. It is one of 480 possible sets of 7 triads (one of two shown in the octonion article) and is the one based on the Cayley–Dickson construction of from two possible quaternion constructions from the . The binary representations of the indices of these triples to 0. These 35 triads are:

{ {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} }

Zero divisors
The list of 84 sets of zero divisors \{e_a, e_b, e_c, e_d\}, where (e_a + e_b) \circ (e_c + e_d) = 0:

\begin{array}{c} \text{Sedenion Zero Divisors} \quad \{ e_a, e_b, e_c, e_d \} \\ \text{where} ~ (e_a + e_b) \circ (e_c + e_d) = 0 \\ \begin{array}{ccc} 1 \leq a \leq 6, & c > a, & 9 \leq b \leq 15 \\ \end{array} \\ \\ \begin{array}{lccr} \{ 9 \leq d \leq 15 \} & \{ -9 \geq d \geq -15 \} & \{ 9 \leq d \leq 15 \} & \{ -9 \geq d \geq -15 \}\\ \end{array} \\ \\ \begin{array}{lccr} \{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_2, -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_7, e_{11}\} & \{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_{12}, e_7, -e_{13}\} \end{array} \end{array}

Space of Zero Divisors
It has been shown that the pairs of zero divisors in the unit sedonions form a manifold isomorphic to the Lie group G2 in the space \mathbb{S}^2. The geometry of sedenion zero divisors Https://arxiv.org/pdf/2411.18881

Applications 
showed that the space of pairs of norm-one sedenions that multiply to zero is [[homeomorphic|homeomorphism]] to the compact form of the exceptional [[Lie group]] G2.  (Note that in his paper, a "zero divisor" means a ''pair'' of elements that multiply to zero.)
     

demonstrated that the three generations of [[lepton]]s and [[quark]]s that are associated with unbroken [[gauge symmetry]] \mathrm {SU(3)_{c} \times U(1)_{em}} can be represented using the algebra of the complexified sedenions \mathbb {C \otimes S}. Their reasoning follows that a primitive [[idempotent]] projector \rho_{+} = 1/2(1+ie_{15}) — where e_{15} is chosen as an [[imaginary unit]] akin to e_{7} for \mathbb {O} in the [[Fano plane]] — that [[acts|Group action]] on the [[standard basis]] of the sedenions uniquely divides the algebra into three sets of split basis elements for \mathbb {C \otimes O}, whose adjoint left actions ''on themselves'' generate three copies of the [[Clifford algebra]] \mathrm Cl(6) which in-turn contain minimal left ideals that describe a single generation of [[fermion]]s with unbroken \mathrm {SU(3)_{c} \times U(1)_{em}} gauge symmetry. In particular, they note that [[tensor product]]s between normed division algebras generate zero divisors akin to those inside \mathbb {S}, where for \mathbb {C \otimes O} the lack of alternativity and associativity does not affect the construction of minimal left ideals since their underlying split basis requires only two basis elements to be multiplied together, in-which associativity or alternativity are uninvolved. Still, these ideals constructed from an adjoint algebra of left actions of the algebra on itself remain associative, alternative, and [[isomorphic]] to a Clifford algebra. Altogether, this permits three copies of (\mathbb {C \otimes O})_{L} \cong \mathrm {Cl(6)} to exist inside \mathbb {(C \otimes S)}_{L}. Furthermore, these three complexified octonion subalgebras are not independent; they share a common \mathrm Cl(2) subalgebra, which the authors note could form a theoretical basis for CKM and PMNS matrices that, respectively, describe [[quark mixing]] and neutrino oscillations.

Sedenion neural networks provide a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.

See also
  • Pfister's sixteen-square identity
  • Split-complex number
  • PG(3,2)

Notes
  • L. S. Saoud and H. Al-Marzouqi, "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm," in IEEE Access, vol. 8, pp. 144823-144838, 2020, .

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