The command AlgebraLibraryData retrieves the structure equations for any of the following real algebras: the real numbers $\mathrm{\ℝ}$, the complex numbers $\mathrm{\ℂ}\mathit{\,}\mathit{ }$the quaternions $\mathrm{\ℍ}\mathit{\,}\mathit{ }$the octonions $\mathrm{\𝕆}\mathit{\,}\mathit{ }$the Clifford algebras $\mathrm{\ℂl}\left(n\, Q\right)$on ${\mathrm{\ℝ}}^n$ with respect to the quadratic form $Q$, and the Jordan algebras $𝕁\left(n\mathit{\,}ℂ\right)\,𝕁\left(n\mathit{\,}ℍ\right)\, 𝕁\left(n\mathit{\,}𝕆\right)$ for small values of $n$.

The keyword argument type ="Split" may be applied to the algebras $\mathrm{\ℂ}\mathit{\,}\mathit{ }\mathrm{\ℍ}\,\mathrm{\𝕆}$ to obtain their split forms. The argument type ="Split" can be applied to $𝕁\left(n\mathit{\,}ℂ\right)\,𝕁\left(n\mathit{\,}ℍ\right)\, 𝕁\left(n\mathit{\,}𝕆\right)$ to obtain the Jordan algebras defined over the split complex numbers, the split quaternions, or the split octonions.

There are two generally accepted versions of the structure equations for the octonions. These are described in Example 2.

The keyword argument quadraticform = Q can be used create the general Clifford algebras, defined with respect to a quadratic form. See Example 3.

For the following small values of $n\,$the structure equations have been stored in Maple and are available without computation: $𝕁\left(n\mathit{\,}\mathrm{\ℝ}\right)$for $n \= 2\, 3\, 4\,5\;$$𝕁\left(n\mathit{\,}\mathrm{\ℂ}\right)$for $n \= 2\, 3\,4\,5\;$$𝕁\left(n\mathit{\,}\mathrm{\ℍ}\right)$for $n \= 2\, 3\, 4\;$$𝕁\left(n\mathit{\,}\mathrm{\𝕆}\right)$for $n \= 2\, 3.$More generally, Jordan algebras can be created using the command JordanMatrices, JordanProduct, and AlgebraData.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{AD1a}≔\mathrm{AlgebraLibraryData}\left(''Quaternions''\,H\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD1a}\,'\left[e\,i\,j\,k\right]'\,'\left[\mathrm{\omega }\right]'\right)$

$\mathrm{algebra\; name:\; H}$

$\mathrm{AD1b}≔\mathrm{AlgebraLibraryData}\left(''Quaternions''\,\mathrm{Hs}\,\mathrm{type}=''Split''\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD1b}\,'\left[e\,i\,j\,k\right]'\,\left[\mathrm{\omega }\right]\right)$

$\mathrm{algebra\; name:\; Hs}$

$\mathrm{MultiplicationTable}\left(H\,''AlgebraTable''\right),\mathrm{MultiplicationTable}\left(\mathrm{Hs}\,''AlgebraTable''\right)$

$\mathrm{AD2a}≔\mathrm{AlgebraLibraryData}\left(''Octonions''\,\mathrm{O1}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD2a}\,'\left[\mathrm{e0}\,\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\,\mathrm{e6}\,\mathrm{e7}\right]'\,\left[\mathrm{\omega }\right]\right)$

$\mathrm{algebra\; name:\; O1}$

$\mathrm{MultiplicationTable}\left(\mathrm{O1}\,''AlgebraTable''\right)$

$\mathrm{AD2b}≔\mathrm{AlgebraLibraryData}\left(''Octonions''\,\mathrm{O2}\,\mathrm{version}=2\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD2b}\,'\left[\mathrm{e0}\,\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\,\mathrm{e6}\,\mathrm{e7}\right]'\,\left[\mathrm{\omega }\right]\right)$

$\mathrm{algebra\; name:\; O2}$

$\mathrm{MultiplicationTable}\left(\mathrm{O2}\,''AlgebraTable''\right)$

$\mathrm{AD3a}≔\mathrm{AlgebraLibraryData}\left(''Clifford(3)''\,\mathrm{Cl3}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD3a}\,'\left[\mathrm{e0}\,\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e12}\,\mathrm{e13}\,\mathrm{e23}\,\mathrm{e123}\right]'\,'\left[\mathrm{\omega }\right]'\right)$

$\mathrm{algebra\; name:\; Cl3}$

$\mathrm{MultiplicationTable}\left(\mathrm{Cl3}\,''AlgebraTable''\right)$

$\mathrm{I12}≔\mathrm{Matrix}\left(\left[\left[1\,0\,0\right]\,\left[0\,-1\,0\right]\,\left[0\,0\,-1\right]\right]\right)$

$\mathrm{AD3b}≔\mathrm{AlgebraLibraryData}\left(''Clifford(3)''\,\mathrm{Cl3Q}\,\mathrm{quadraticform}=\mathrm{I12}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD3b}\,'\left[\mathrm{e0}\,\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e12}\,\mathrm{e13}\,\mathrm{e23}\,\mathrm{e123}\right]'\,'\left[\mathrm{\omega }\right]'\right)$

$\mathrm{algebra\; name:\; Cl3Q}$

$\mathrm{MultiplicationTable}\left(\mathrm{Cl3Q}\,''AlgebraTable''\right)$

$\mathrm{AD3b}≔\mathrm{AlgebraLibraryData}\left(''Clifford(2)''\,\mathrm{Cl2}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD3b}\,'\left[\mathrm{e0}\,\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]'\,'\left[\mathrm{\omega }\right]'\right)$

$\mathrm{algebra\; name:\; Cl2}$

$\mathrm{MultiplicationTable}\left(H\,''AlgebraTable''\right),\mathrm{MultiplicationTable}\left(\mathrm{Cl2}\,''AlgebraTable''\right)$

$\mathrm{AD4}≔\mathrm{AlgebraLibraryData}\left(''Jordan(2,\; Quaternions)''\,\mathrm{J2H}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{AD4}\right)$

$\mathrm{algebra\; name:\; J2H}$

$\mathrm{MultiplicationTable}\left(\mathrm{J2H}\,''AlgebraTable''\right)$