The structure of certain \(K_2O_F\) (Q2721532)

Let \(O_F=O(d)\) denote the ring of integers in the quadratic number field \(F={\mathbb{Q}}\sqrt{d}\). The author finds an element of order 3 in \(K_2 O(-21)\), gives the generator for \(K_2 O(15)\) \(\cong \mathbb{Z}/(2)\oplus \mathbb{Z}/(8) \oplus \mathbb{Z}/(3)\),\ i.e. \(\{-1, -1\}\) and \(\{-3, 3+\sqrt{15}\}\) generate \(\mathbb{Z}/(2)\) and \(\mathbb{Z}/(8)\oplus \mathbb{Z}/(3)\), respectively, and gives the generator for \(K_2 O(29)\) \(\cong \mathbb{Z}/(2)\oplus \mathbb{Z}/(2)\oplus \mathbb{Z}/(3)\), i.e., \(\{-1, (5+\sqrt{29})/2\}\) and \(\{(1+\sqrt{29})/2, 8\}\) generate \(\mathbb{Z}/(2)\oplus \mathbb{Z}/(2)\) and \(\mathbb{Z}/(3)\) respectively.