Editing Subalgebra (section)

=== Example ===

The 2×2-matrices over the reals '''R''', with [[matrix multiplication]], form a four-dimensional unital algebra M(2,'''R'''). The 2×2-matrices for which all entries are zero, except for the first one on the diagonal, form a subalgebra. It is also unital, but it is not a unital subalgebra.

The [[identity element]] of M(2,'''R''') is the [[identity matrix]] I , so the unital subalgebras contain the line of [[diagonal matrices]] {''x'' I  : ''x'' in '''R'''}. For two-dimensional subalgebras, consider
:<math>E^2 = \begin{pmatrix}a & c \\ b & -a \end{pmatrix}^2 = \begin{pmatrix}a^2+bc & 0 \\ 0 & bc+a^2 \end{pmatrix} = p I \ \ \text{where}\ \ p = a^2 + bc .</math>
When ''p'' = 0, then E is [[nilpotent]] and the subalgebra { ''x'' I + ''y'' E : ''x, y'' in '''R''' } is a copy of the [[dual number]] plane. When ''p'' is negative, take ''q'' = 1/√−''p'', so that (''q'' E)<sup>2</sup> = &minus; I, and subalgebra { ''x'' I + ''y'' (''q''E) : ''x,y'' in '''R''' } is a copy  of the [[complex plane]]. Finally, when ''p'' is positive, take ''q'' = 1/√''p'', so that (''q''E)<sup>2</sup> = I, and subalgebra { ''x'' I + ''y'' (''q''E) : ''x,y'' in '''R''' } is a copy  of the plane of [[split-complex number]]s. By the [[law of trichotomy]], these are the only planar subalgebras of M(2,'''R''').

[[L. E. Dickson]] noted in 1914, the "Equivalence of [[complex quaternion]] and complex matric algebras", meaning M(2,'''C'''), the 2x2 complex matrices.<ref>[[L. E. Dickson]] (1914) ''Linear Algebras'', pages 13,4</ref> But he notes also, "the real quaternion and real matric sub-algebras are not [isomorphic]." The difference is evident as there are the three [[isomorphism class]]es of planar subalgebras of M(2,'''R'''), while real quaternions have only one isomorphism class of planar subalgebras as they are all isomorphic to '''C'''.