Conditional Multivariate Gaussian Distribution - Section 2.3, equation 2.74
\begin{align}&amp;-\frac12(x - \mu)^{T}\Sigma^{-1}(x - \mu) \\&amp;= -\frac12(x_a - \mu_a)^{T}\Lambda_{aa}(x_a - \mu_a) \tag{1}
\\&amp;-\frac12(x_a - \mu_a)^{T}\Lambda_{ab}(x_b - \mu_b) \tag{2}
\\&amp;-\frac12(x_b - \mu_b)^{T}\Lambda_{ba}(x_a - \mu_a) \tag{3}
\\&amp;-\frac12(x_b - \mu_b)^{T}\Lambda_{aa}(x_b - \mu_b) \tag{4}\end{align}

Now let's focus on $(1)$,  the linear part in $x_a$ are 

$$-\frac12 (-x_a^T\Lambda_{aa}\mu_a - \mu_a^T\Lambda_{aa}x_a)=x_a^T\Lambda_{aa}\mu_a \tag{5}$$

Now focus on $(2)$, the linear part in $x_a$ is 

$$-\frac12x_a^T\Lambda_{ab}(x_b-\mu_b) \tag{6}$$

Now focus on $(3)$,  the linear part in $x_a$ is 

$$-\frac12(x_b-\mu_b)^T\Lambda_{ba}x_a=-\frac12x_a^T\Lambda_{ab}(x_b-\mu_b) \tag{7}$$

There is no linear term of $x_a$ in $(4)$, 

Adding $(5)$ to $(7)$, we have 

\begin{align}x_a^T\Lambda_{aa}\mu_a-\frac12x_a^T\Lambda_{ab}(x_b - \mu_b)-\frac12x_a^T\Lambda_{ab}(x_b - \mu_b)&amp;=x_a^T\Lambda_{aa}\mu_a-x_a^T\Lambda_{ab}(x_b - \mu_b)\\
&amp;=x_a^T(\Lambda_{aa}\mu_a -\Lambda_{ab}(x_b-\mu_b))\end{align}