Special factors of invertible elements in simple unital purely infinite C*-algebras

In simple unital purely infinite C*-algebra A, Leen proved that any element in the identity component of the invertible group is a finite product of symmetries of A. Revising Leen's factorization, we show that a multiple of eight of such factors are *-symmetries of the form 1-2P _i,j (u), where P _i,j (u) are certain projections of the C*-matrix algebra, defined by Dye as P _i,j (u) = 1/2 (e _i,i + e _j,j + e _i,1 ue _1,j + e _j,1 u*e _1,i ); for a given system of matrix units (e _i,j ) _i,j=1 ⁿ of A and a unitary u ∈ U(A).