The following is my best guess, after reading the book.

The relative covariance factor, $\Lambda_\theta$, is a $q \times q$ matrix (dimensions are explained in the excerpt you posted).

  For a model with a simple scalar random-effects term, (p. 15, Fig. 1.3) it is calculated as a multiple of $\theta$ and identity matrix of dimensions $q \times q$:

$$\Lambda_\theta = \theta \times {I_q}$$

This is the general way to calculate $\Lambda_\theta$, and it is modified according to the number of random-effects and their covariance structure. For a model with two uncorrelated random-effects terms in a crossed design, as on pp. 32-34, it is block diagonal with two blocks each of which is a multiple of $\theta$ and identity (p. 34, Fig. 2.4):

Same with two nested random-effects terms (p. 43, Fig. 2.10, not shown here).

For a longitudinal (repeated-measures) model with a random intercept and a random slope which are allowed to correlate $\Lambda_\theta$ consists of triangular blocks representing both random-effects and their correlation (p. 62, Fig. 3.2):

Modelling the same dataset with two uncorrelated random-effects terms (p. 65, Fig. 3.3) returns $\Lambda_\theta$ of the same structure as shown previously, in Fig. 2.4:

Additional notes:

The variance-component parameter vector $\theta$ is estimated iteratively to minimise the model deviance $\widetilde{d}$ according to eq. 1.10 (p. 14).

$\theta_i = \frac{\sigma_i}{\sigma}$ Where $\sigma_i$ refers to the square root of the i-th random-effect variance, and $\sigma$ refers to the square root of the residual variance (compare with pp. 32-34).

The book version from June 25, 2010 refers to a version of lme4 which has been modified. One of the consequences is that in the current version 1.1.-10. the random-effects model object-class merMod has a different structure and $\Lambda_\theta$ is accessed in a different way, using the method getME:

image(getME(fm01ML, "Lambda"))

The variance-component parameter vector $\theta$ is estimated iteratively to minimise the model deviance $\widetilde{d}$ according to eq. 1.10 (p. 14).

The relative covariance factor, $\Lambda_\theta$, is a $q \times q$ matrix (dimensions are explained in the excerpt you posted). For a model with a simple scalar random-effects term, (p. 15, Fig. 1.3) it is calculated as a multiple of $\theta$ and identity matrix of dimensions $q \times q$:

$$\Lambda_\theta = \theta \times {I_q}$$

This is the general way to calculate $\Lambda_\theta$, and it is modified according to the number of random-effects and their covariance structure. For a model with two uncorrelated random-effects terms in a crossed design, as on pp. 32-34, it is block diagonal with two blocks each of which is a multiple of $\theta$ and identity (p. 34, Fig. 2.4):

Same with two nested random-effects terms (p. 43, Fig. 2.10, not shown here).

For a longitudinal (repeated-measures) model with a random intercept and a random slope which are allowed to correlate $\Lambda_\theta$ consists of triangular blocks representing both random-effects and their correlation (p. 62, Fig. 3.2):

Modelling the same dataset with two uncorrelated random-effects terms (p. 65, Fig. 3.3) returns $\Lambda_\theta$ of the same structure as shown previously, in Fig. 2.4:

Additional notes:

$\theta_i = \frac{\sigma_i}{\sigma}$ Where $\sigma_i$ refers to the square root of the i-th random-effect variance, and $\sigma$ refers to the square root of the residual variance (compare with pp. 32-34).

The book version from June 25, 2010 refers to a version of lme4 which has been modified. One of the consequences is that in the current version 1.1.-10. the random-effects model object-class merMod has a different structure and $\Lambda_\theta$ is accessed in a different way, using the method getME:

image(getME(fm01ML, "Lambda"))

The following is my best guess, after reading the book.

The relative covariance factor, $\Lambda_\theta$, is a $q \times q$ matrix (dimensions are explained in the excerpt you posted).

For a model with a simple scalar random-effects term, (p. 15, Fig. 1.3) it is calculated as a multiple of $\theta$ and identity matrix of dimensions $q \times q$:

$$\Lambda_\theta = \theta \times {I_q}$$

This is the general way to calculate $\Lambda_\theta$, and it is modified according to the number of random-effects and their covariance structure. For a model with two uncorrelated random-effects terms in a crossed design, as on pp. 32-34, it is block diagonal with two blocks each of which is a multiple of $\theta$ and identity (p. 34, Fig. 2.4):

Same with two nested random-effects terms (p. 43, Fig. 2.10, not shown here).

For a longitudinal (repeated-measures) model with a random intercept and a random slope which are allowed to correlate $\Lambda_\theta$ consists of triangular blocks representing both random-effects and their correlation (p. 62, Fig. 3.2):

Modelling the same dataset with two uncorrelated random-effects terms (p. 65, Fig. 3.3) returns $\Lambda_\theta$ of the same structure as shown previously, in Fig. 2.4:

Additional notes:

The variance-component parameter vector $\theta$ is estimated iteratively according to minimise the model deviance according to eq. 1.10 (p. 14).

$\theta_i = \frac{\sigma_i}{\sigma}$ Where $\sigma_i$ refers to the square root of the i-th random-effect variance, and $\sigma$ refers to the square root of the residual variance (compare with pp. 32-34).

The book version from June 25, 2010 refers to a version of lme4 which has been modified. One of the consequences is that in the current version 1.1.-10. the random-effects model object-class merMod has a different structure and $\Lambda_\theta$ is accessed in a different way, using the method getME:

image(getME(fm01ML, "Lambda"))

The following is my best guess, after reading the book.

The relative covariance factor, $\Lambda_\theta$, is a $q \times q$ matrix (dimensions are explained in the excerpt you posted).

For a model with a simple scalar random-effects term, (p. 15, Fig. 1.3) it is calculated as a multiple of $\theta$ and identity matrix of dimensions $q \times q$:

$$\Lambda_\theta = \theta \times {I_q}$$

This is the general way to calculate $\Lambda_\theta$, and it is modified according to the number of random-effects and their covariance structure. For a model with two uncorrelated random-effects terms in a crossed design, as on pp. 32-34, it is block diagonal with two blocks each of which is a multiple of $\theta$ and identity (p. 34, Fig. 2.4):

Same with two nested random-effects terms (p. 43, Fig. 2.10, not shown here).

For a longitudinal (repeated-measures) model with a random intercept and a random slope which are allowed to correlate $\Lambda_\theta$ consists of triangular blocks representing both random-effects and their correlation (p. 62, Fig. 3.2):

Modelling the same dataset with two uncorrelated random-effects terms (p. 65, Fig. 3.3) returns $\Lambda_\theta$ of the same structure as shown previously, in Fig. 2.4:

Additional notes:

The variance-component parameter vector $\theta$ is estimated iteratively to minimise the model deviance $\widetilde{d}$ according to eq. 1.10 (p. 14).

$\theta_i = \frac{\sigma_i}{\sigma}$ Where $\sigma_i$ refers to the square root of the i-th random-effect variance, and $\sigma$ refers to the square root of the residual variance (compare with pp. 32-34).

The book version from June 25, 2010 refers to a version of lme4 which has been modified. One of the consequences is that in the current version 1.1.-10. the random-effects model object-class merMod has a different structure and $\Lambda_\theta$ is accessed in a different way, using the method getME:

image(getME(fm01ML, "Lambda"))