Similar arguments about the Hilbert inner product ( which can be demonstrated to be a Hermitian form, therefore justifying the name " inner product " ) lead to the conclusion that its neutral space is precisely K _ { 00 } = ( K _ 0 \ cap K _ + ) \ oplus ( K _ 0 \ cap K _-), that elements of this neutral space have zero Hilbert inner product with any element of K, and that the Hilbert inner product is positive semi-definite.
32.
There is a division algebra " D " with center " l " and degree over " l " 3 or 1, with an involution of the second kind which restricts to the nontrivial automorphism of " l " over " k ", and a nontrivial Hermitian form on a module over " D " of dimension 1 or 3 such that " G " is the special unitary group of this Hermitian form . ( As a consequence of and the work of Cartwright and Steger, " D " has degree 3 over " l " and the module has dimension 1 over " D " . ) There is one real place of " k " such that the points of " G " form a copy of PU ( 2, 1 ), and over all other real places of " k " they form the compact group PU ( 3 ).
33.
There is a division algebra " D " with center " l " and degree over " l " 3 or 1, with an involution of the second kind which restricts to the nontrivial automorphism of " l " over " k ", and a nontrivial Hermitian form on a module over " D " of dimension 1 or 3 such that " G " is the special unitary group of this Hermitian form . ( As a consequence of and the work of Cartwright and Steger, " D " has degree 3 over " l " and the module has dimension 1 over " D " . ) There is one real place of " k " such that the points of " G " form a copy of PU ( 2, 1 ), and over all other real places of " k " they form the compact group PU ( 3 ).
