I'm stuck on how to go from the first line to the second line in this equation related to the Gauss-Markov model where $\mathbf{y}=X\mathbf{b}+\mathbf{e}$, $E(\mathbf{e})=0$, and $Cov(\mathbf{e})=\sigma^2I_n$: \begin{align} Cov(\Lambda^T \mathbf{\hat b}) =Cov(\Lambda^T (X^TX)^g X^T y) &= \sigma^2 \Lambda^T (X^TX)^gX^TX(X^TX)^{g^T} \Lambda \\ &=\sigma^2\Lambda^T(X^TX)^g\Lambda \end{align}
In the equation the columns $\lambda^{(j)}$ of the matrix $\Lambda$ are linear independent such that $\lambda^{(j)T} \mathbf{b}$ are linearly independent estimable functions. $X$ is the design matrix and $(X^TX)^g$ is the generalized inverse of $X^TX$.
An explanation of how to simply $\sigma^2 \Lambda^T (X^TX)^gX^TX(X^TX)^{g^T} \Lambda$ to $\sigma^2\Lambda^T(X^TX)^g\Lambda$ would be most appreciated.
For reference this is on p. 75 of A Primer on Linear Models by John F. Monahan.
Thanks in advance for your help --
It suffices to show that $X^{T}X (X^{T}X)^{g^{T}}$ is the identity matrix, which is true because you can interchange the generalized inverse and matrix transpose. Then the result follows by noticing that $X^{T}X$ is a symmetric matrix.