We give exhaustive list of biquadratic fields $K=\mathbb{Q}(i,\sqrt{m})$ and $K=\mathbb{Q}(\sqrt{2},\sqrt{m})$ without $2$-exotic symbol, i.e. for which the $2$-rank of the Hilbert kernel (or wild kernel) is zero. Such $K=\mathbb{Q}(i,\sqrt{m})$ are logarithmic principals [J3]. We detail an exemple of this technical numerical exploration and quote the family of theories and results we utilize. The $2$-rank of tame, regular and wild kernel of $K$-theory are connected with local and global problem of embedding in a $Z_2$-extension. Global class field theory can describe the $2$-rank of the Hilbert...