Suppose (X1,...,Xk) has the multivariate normal distribution with unknown mean vector = (1,...,k) and known covariance

Suppose (X1,...,Xk) has the multivariate normal distribution with unknown mean vector ξ = (ξ1,...,ξk) and known covariance matrix Σ.

Suppose X1 is independent of (X2,...,Xk). Show that X1 is partially sufficient for ξ1 in the sense of Problem 3.60. Provide an alternative argument for Case 2 of Example 3.9.2.