  The H\"usler-Reiss distribution describes the limit of the pointwise maxima
of a bivariate normal distribution. This distribution is defined by a single
parameter, $\lambda$. We provide asymptotic theory for maximum likelihood
estimation of $\lambda$ under a block maxima approach. Our work assumes
independent and identically distributed bivariate normal random variables,
grouped into blocks where the block size and number of blocks increase
simultaneously. With these assumptions our results provide conditions for the
asymptotic normality of the Maximum Likelihood Estimator (MLE). We characterize
the bias of the MLE, provide conditions under which this bias is asymptotically
negligible, and discuss how to choose the block size to minimize a
bias-variance trade-off. The proofs are an extension of previous results for
choosing the block size in the estimation of univariate extreme value
distributions (Dombry and Ferreria 2019), providing a potential basis for
extensions to multivariate cases where both the marginal and dependence
parameters are unknown. The proofs rely on the Argmax Theorem applied to a
localized loglikelihood function, combined with a Lindeberg-Feller Central
Limit Theorem argument to establish asymptotic normality. Possible applications
of the method include composite likelihood estimation in Brown-Resnick
processes, where it is known that the bivariate distributions are of
H\"usler-Reiss form.
