who pioneered the study of the sedenions? The nature of this question is pure historical curiosity.
I found lots of background information about the discovery of both Imaginary and Complex Numbers, and enough information about the first two types of Hyper Complex Numbers; Quaternions and Octonions (also known as Cayley Numbers).
However, I haven't found who were the mathematicians that pioneered the study of the sedenions.
It is known that this can be achieved today through the Cayley-Dickson doubling process.
This question is not about the usefulness of the sedenions, or the operations that are allowed on them, but who pioneered their study and when. This is a historical question.
 A: On Bibliography of Quaternions and Allied Mathematics by Alexander Macfarlane I found this:
On page 72; James Byrnie Shaw
1896 Sedenions (title). American Assoc. Proc., 45, 26.
I couldn't find this reference, but the same author wrote this book:
Synopsis of Linear Associative Algebra: A Report on its Natural Development and Results Reached up to the Present Time. 1907.
From its Table of Contents; Part II: Particular Algebras. Section XVIII: Triquaternions and Quadriquaternions. Page 91.
Early on Sedenions were also known as "quadriquaternions".
Section XIX: Sylvester Algebras. Page 93.
Covers "Nonions" (9-ions), and "Sedenions" (16-ions).
Here the Sedenions are attributed to James Joseph Sylvester.
On page 76; James Joseph Sylvester
1883-4 On quaternions, nonions, sedenions, etc. Johns Hopkins
Univ. Circ., 3. Nos. 7 and 9. 4, No. 28.
This second reference can be found among The Collected Mathematical Papers of James Joseph Sylvester, [Volume IV(1882—1897)]:
Sylvester, James Joseph (1973) [1904], Baker, Henry Frederick (ed.), The collected mathematical papers of James Joseph Sylvester, IV, New York: AMS Chelsea Publishing, ISBN 978-0-8218-4238-6
Then, as far as I understood, James Joseph Sylvester appears on the literature as the proponent of two Algebras; the one from the Nonions and the Sedenions. I will leave as an open question for the community to confirm, correct, or debunk that J.J. Sylvester "discovered" or "was the pioneer" of their study in 1883.
A: Here is an early reference. The term sedenion is not used in this paper, but the Cayley-Dickson process is used to produce algebras with dimension a power of $2$ (including 16).
R. D. Schafer, On the algebras formed by the Cayley-Dickson process, Amer. J. Math. 76 (1954), 435-446.
