Let $$ A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, B = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, C = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} $$ be a basis of the Lie algebra $\mathfrak{su}(2)$. Also define $g_\lambda$, for $\lambda > 0$, to be the Riemannian metric on the Lie group $\operatorname{SU}(2)$ such that $(A, B, \lambda C)$ is an orthonormal basis. I want to show that the well known diffeomorphism $$ (z,w)\in\mathbb{S}^3 \mapsto \begin{pmatrix} z & w \\ -\bar w & \bar z \end{pmatrix} \in \operatorname{SU}(2) $$ is an isometry when $\lambda=1$ and $\mathbb{S}^3$ is equipped with its standard round metric.
I tried to use charts on both sides but it's too much calculus and doesn't seem to go somewhere.