In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface S or blackboard bold

The sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mathematically expressed as

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 trigintaduonions or sometimes the 32-nions.

[2] The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matrices over the real numbers, or that studied by Smith (1995).

, which form a basis of the vector space of sedenions.

Like octonions, 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.

They do, however, have the property of power associativity, which can be stated as that, for any element

The sedenions have a multiplicative identity element

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

All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

The particular sedenion multiplication table shown above is represented by 35 triads.

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 quaternions from two possible quaternion constructions from the complex numbers.

The binary representations of the indices of these triples bitwise XOR 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} } The list of 84 sets of zero divisors

Moreno (1998) showed that the space of pairs of norm-one sedenions that multiply to zero is homeomorphic 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.)

Guillard & Gresnigt (2019) demonstrated that the three generations of leptons and quarks that are associated with unbroken gauge symmetry

can be represented using the algebra of the complexified sedenions

Their reasoning follows that a primitive idempotent projector

is chosen as an imaginary unit akin to

in the Fano plane — that acts on the standard basis of the sedenions uniquely divides the algebra into three sets of split basis elements for

, whose adjoint left actions on themselves generate three copies of the Clifford algebra

which in-turn contain minimal left ideals that describe a single generation of fermions with unbroken

In particular, they note that tensor products between normed division algebras generate zero divisors akin to those inside

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.

Furthermore, these three complexified octonion subalgebras are not independent; they share a common

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[further explanation needed] a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.